Rewrite the integral (Equation 5.9.1) in terms of u: ∫(x2 − 3)3(2xdx) = ∫u3du. Using the power rule for integrals, we have. ∫u3du = u4 4 + C. Substitute the original expression for x back into the solution: u4 4 + C = (x2 − 3)4 4 + C. We can generalize the procedure in the following Problem-Solving Strategy.Lesson 16: Trigonometric substitution. Introduction to trigonometric substitution. Substitution with x=sin (theta) More trig sub practice. Trig and u substitution together (part 1) Trig and u substitution together (part 2) Trig substitution with tangent. More trig substitution with tangent. Long trig sub problem.Solve the integral of sec(x) by using the integration technique known as substitution. The technique is derived from the chain rule used in differentiation. The problem requires a ...The objective of Integration by substitution is to substitute the integrand from an expression with variable to an expression with variable where = Theory We want to transform ... Substitute back the values for u for indefinite integrals. 6. Don't forget the constant of integration for indefinite integrals. Finding u ...Translate all your x's into u's everywhere in the integral, including the dx. When you're done, you should have a new integral that is entirely in u. If you ...We use the substitution x = sinu to transform the function from x2√1 − x2 to sin2u√1 − sin2u, and we also convert dx to cosudu. Finally, we convert the limits 0 and 1 to 0 and π / 2. This transforms the integral in Equation 15.7.1: ∫1 0x2√1 − x2dx = ∫π / …Jan 22, 2020 · Turning the Tables on Tough Integrals. In our previous lesson, Fundamental Theorem of Calculus, we explored the properties of Integration, how to evaluate a definite integral (FTC #1), and also how to take a derivative of an integral (FTC #2). In this lesson, we will learn U-Substitution, also known as integration by substitution or simply u ... May 7, 2018 · With the basics of integration down, it's now time to learn about more complicated integration techniques! We need special techniques because integration is ... u-substitution-integration-calculator. en. Related Symbolab blog posts. High School Math Solutions – Partial Fractions Calculator. Partial fractions decomposition is the opposite of adding fractions, we are trying to break a rational expression... Read More. Enter a problem. Cooking Calculators.Explore math with our beautiful, free online graphing calculator. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more.In the same way that log_10(1000) = 3 means that “the power that 10 is raised to to equal 1000 is 3”, ln 2 means “the power that e is raised to to equal 2”. So ...Alliance Integrated Metaliks News: This is the News-site for the company Alliance Integrated Metaliks on Markets Insider Indices Commodities Currencies StocksU Substitution Formula: The technique known as U-substitution, or integration by substitution in calculus, provides a method for solving integrals. It stands as a crucial method in mathematics due to its relation to the fundamental theorem of calculus, which is typically used for finding antiderivatives. The U-substitution formula …In summary, the conversation discusses the solution to a problem involving integration and u substitution, specifically the integrals 1/(8-4x) and 1/(2x). The solution involves rewriting the integrals algebraically and using u substitution to simplify them.To understand integration by substitution, you can just use the chain rule in reverse: \begin{equation} \int f(g (x)) g'(x) dx = F (g (x)) + C, \end{equation} where $ F $ is an anti derivative of $ f $. To check this, just take the derivative of …Jan 29, 2022 · What Is U-Substitution. You’re probably familiar with the idea that integration is the reverse process of differentiation. U-substitution is an integration technique that specifically reverses the chain rule for differentiation. Because of this, it’s common to refer to u-substitution as the reverse chain rule. Now we're almost there: since \( u=1-x^2\), \( x^2=1-u\) and the integral is \[\int -{1\over2}(1-u)\sqrt{u}\,du.\] It's no coincidence that this is exactly the integral we computed in (8.1.4), …Here is a summary for the sine trig substitution. √a2 − b2x2 ⇒ x = a bsinθ, − π 2 ≤ θ ≤ π 2. There is one final case that we need to look at. The next integral will also contain something that we need to make sure we can deal with. Example 5 Evaluate the following integral. ∫ 1 60 x5 (36x2 + 1)3 2 dx. Show Solution.Dec 28, 2012 · Courses on Khan Academy are always 100% free. Start practicing—and saving your progress—now: https://www.khanacademy.org/math/ap-calculus-ab/ab-integration-... 20 hours ago · The U-Substitution Calculator is a powerful tool in calculus, simplifying integration through the U-substitution. This digital calculator allows users to input complex integral expressions and systematically guides them through the steps of u-substitution. The calculator converts the integral into a more manageable form by selecting an ...Rewrite the integral (Equation 5.5.1) in terms of u: ∫(x2 − 3)3(2xdx) = ∫u3du. Using the power rule for integrals, we have. ∫u3du = u4 4 + C. Substitute the original expression for x back into the solution: u4 4 + C = (x2 − 3)4 4 + C. We can generalize the procedure in the following Problem-Solving Strategy.So, an effective strategy is to break the function into two pieces that are multiplied by each other. The first piece will be something you can use u substitution for to get the integral into a form you know how to integrate. The second piece will be the derivative of whatever you called u. If you cannot do that, you cannot use u substitution. Aug 25, 2018 · MIT grad shows how to do integration using u-substitution (Calculus). To skip ahead: 1) for a BASIC example where your du gives you exactly the expression yo... Nov 9, 2022 · Example 5.3.1. Determine the general antiderivative of. h(x) = (5x − 3)6. Check the result by differentiating. For this composite function, the outer function f is f(u) = u6, while the inner function is u(x) = 5x − 3. Since the antiderivative of f is F(u) = 1 7u7 + C, we see that the antiderivative of h is. What do you do if a recipe calls for baking soda but you only have baking powder, or if you have baking soda but not baking powder? As it turns out, there are options. You can make...So this is all equal to negative 243 times the indefinite integral of u squared minus u to the fourth-- I'm just distributing the u squared-- du. Now, this is ...Use substitution to replace \(x \to u\) and \(dx \to du\), and cancel any remaining \(x\) terms if possible. Integrate with respect to \(u\). If at this point you still have any \(x\)s in your …Course: AP®︎/College Calculus AB > Unit 6. Lesson 11: Integrating using substitution. 𝘶-substitution intro. 𝘶-substitution: multiplying by a constant. 𝘶-substitution: defining 𝘶. 𝘶-substitution: defining 𝘶 (more examples) 𝘶-substitution. 𝘶-substitution: definite integral of exponential function. Math >. Explore math with our beautiful, free online graphing calculator. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more.Oct 19, 2021 · u u -substitution. Find the indefinite integral ∫ 8(ln(x))3 x dx ∫ 8 ( ln ( x)) 3 x d x. Again, we will go through the steps of u u -substitution. The inside function in this case is ln(x) ln. . ( x). We can see that the derivative is 1 x 1 x, and this is good since there is an x x dividing the rest of the problem. Trigonometric substitution is a technique of integration that involves replacing the original variable by a trigonometric function. This can help to simplify integrals that contain expressions like a^2 - x^2, a^2 + x^2, or x^2 - a^2. In this section, you will learn how to apply this method and how to choose the appropriate substitution for different …Nimble, a global leader in providing simple and smart CRM for small business teams, has announced a new CRM integration with Microsoft Teams. Nimble, a global leader in providing s...A heart-healthy diet is low in saturated fat. Saturated fat can increase your bad cholesterol and clog your arteries. A heart-healthy diet also limits foods with added salt, which ...This means ∫π0sin(x)dx = ( − cos(π)) − ( − cos(0)) = 2. Sometimes an approximation to a definite integral is desired. A common way to do so is to place thin rectangles under the curve and add the signed areas together. Wolfram|Alpha …Do you want to learn how to integrate functions using u-substitution? This pdf file from Illinois Institute of Technology explains the method step by step with ...The definite integral of from to , denoted , is defined to be the signed area between and the axis, from to . Both types of integrals are tied together by the fundamental theorem of calculus. This states that if is continuous on and is its continuous indefinite integral, then . This means . Sometimes an approximation to a definite integral is ...7.2: Trigonometric Integrals Trigonometric substitution is an integration technique that allows us to convert algebraic expressions that we may not be able to integrate into expressions involving trigonometric functions, which we may be able to integrate using the techniques described in this section. In addition, these types of integrals ...Integration by substitution. In calculus, integration by substitution, also known as u-substitution, reverse chain rule or change of variables, [1] is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation, and can loosely be thought of as using the chain rule "backwards." Integration by U substitution, step by step, example. For more free calculus videos visit http://MathMeeting.com.Lesson 16: Trigonometric substitution. Introduction to trigonometric substitution. Substitution with x=sin (theta) More trig sub practice. Trig and u substitution together (part 1) Trig and u substitution together (part 2) Trig substitution with tangent. More trig substitution with tangent. Long trig sub problem.U-substitution in definite integrals is just like substitution in indefinite integrals except that, since the variable is changed, the limits of integration must be changed as well. If you don’t change the limits of integration, then you’ll need to back-substitute for the original variable at the en.Joe Foster u-Substitution Recall the substitution rule from MATH 141 (see page 241 in the textbook). Theorem If u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then ˆ f(g(x))g′(x)dx = ˆ f(u)du. This method of integration is helpful in reversing the chain rule (Can you see why?)Course: AP®︎/College Calculus AB > Unit 6. Lesson 11: Integrating using substitution. 𝘶-substitution intro. 𝘶-substitution: multiplying by a constant. 𝘶-substitution: defining 𝘶. 𝘶-substitution: defining 𝘶 (more examples) 𝘶-substitution. 𝘶-substitution: definite integral of exponential function. Math >. U Substitution for Definite Integrals. In general, a definite integral is a good candidate for u substitution if the equation contains both a function and that function’s derivative. When evaluating definite integrals, figure out the indefinite integral first and then evaluate for the given limits of integration. Example problem: Evaluate:Aug 27, 2018 · GET STARTED. U-substitution to solve integrals. U-substitution is a great way to transform an integral. Finding derivatives of elementary functions was a relatively simple process, because taking the derivative only meant applying the right derivative rules. This is not the case with integration. Unlike derivatives, it may not be immediately ... Oct 20, 2020 · After the substitution, u is the variable of integration, not x. But the limits have not yet been put in terms of u, and this is essential. 4. (nothing to do) u = x ³−5. x = −1 gives u = −6; x = 1 gives u = −4. 5. The integrand still contains x (in the form x ³). Use the equation from step 1, u = x ³−5, and solve for x ³ = u +5. Mar 17, 2022 ... U substitution (Integration by Substitution) is a common approach to solving integrals that contain a composition of functions.The other way, which Sal used here, is to treat it as an indefinite integral (no boundaries) when you do the u-substitution, but then after integrating, transform the result back from u to x. When you do that, you can evaluate the integral in terms of the original boundaries, because you've reversed the effect of the substitution. Use substitution to replace \(x \to u\) and \(dx \to du\), and cancel any remaining \(x\) terms if possible. Integrate with respect to \(u\). If at this point you still have any \(x\)s in your …Options. The Integral Calculator lets you calculate integrals and antiderivatives of functions online — for free! Our calculator allows you to check your solutions to calculus exercises. …Nov 10, 2022 · the \(u\)-substitution \(u = x^2\) is no longer possible because the factor of \(x\) is missing. Hence, part of the lesson of \(u\)-substitution is just how specialized the process is: it only applies to situations where, up to a missing constant, the integrand is the result of applying the Chain Rule to a different, related function.If you have a right triangle with hypotenuse of length a and one side of length x, then: x^2 + y^2 = a^2 <- Pythagorean theorem. where x is one side of the right triangle, y is the other side, and a is the hypotenuse. So anytime you have an expression in the form a^2 - x^2, you should think of trig substitution.U-Substitution Integration, or U-Sub Integration, is the opposite of the The Chain Rule from Differential Calculus, but it’s a little trickier since you have to set it up like a puzzle. Once you get the hang of it, it’s fun, though! U-sub is also known the reverse chain rule or change of variables. One way we can try to integrate is by u -substitution. Let's look at an example: Example 1: Evaluate the integral: Something to notice about this integral is that it consists of both a function f ( x2 +5) and the derivative of that function, f ' (2 x ). This can be a but unwieldy to integrate, so we can substitute a variable in. Lesson 16: Trigonometric substitution. Introduction to trigonometric substitution. Substitution with x=sin (theta) More trig sub practice. Trig and u substitution together (part 1) Trig and u substitution together (part 2) Trig substitution with tangent. More trig substitution with tangent. Long trig sub problem.Feb 1, 2022 ... Examples of using the substitution rule (u-substitution) to evaluate indefinite and definite integrals. Lots of examples to help you ...Integral Calculus (2017 edition) 12 units · 88 skills. Unit 1 Definite integrals introduction. Unit 2 Riemann sums. Unit 3 Fundamental theorem of calculus. Unit 4 Indefinite integrals. Unit 5 Definite integral evaluation. Unit 6 Integration techniques. Unit 7 Area & arc length using calculus. Unit 8 Integration applications.Identifying which function to take as 'u' simply comes with experience. Some integrals like sin (x)cos (x)dx have an easy u-substitution (u = sin (x) or cos (x)) as the 'u' and the derivative are explicitly given. Some like 1/sqrt (x - 9) require a trigonometric ratio to be 'u'. Some other questions make you come up with a completely (seemingly ... May 7, 2018 · With the basics of integration down, it's now time to learn about more complicated integration techniques! We need special techniques because integration is ... This problem exemplifies the situation where we sometimes use both u-substitution and Integration by Parts in a single problem. If we write t 3 = t · t 2 and consider the indefinite integral Z t · t 2 · sin(t 2 ) dt, we can use a mix of the two techniques we have recently learned. First, let z = t 2 so that dz = 2t dt, and thus t dt = 1 2 dz.Sep 26, 2014 · One way we can try to integrate is by u -substitution. Let's look at an example: Example 1: Evaluate the integral: Something to notice about this integral is that it consists of both a function f ( x2 +5) and the derivative of that function, f ' (2 x ). This can be a but unwieldy to integrate, so we can substitute a variable in.The second integral is more difficult because the first integral is simply a \(u\)-substitution type. 7.3: Trigonometric Substitution. Simplify the expressions in exercises 1 - 5 by writing each one using a single trigonometric function. 1) \(4−4\sin^2θ\) 2) \(9\sec^2θ−9\) AnswerSymbolab is the best integral calculator solving indefinite integrals, definite integrals, improper integrals, double integrals, triple integrals, multiple integrals, antiderivatives, and more. Do you want to learn how to integrate functions using u-substitution? This pdf file from Illinois Institute of Technology explains the method step by step with ...In calculus, the integration by substitution method is also known as the “Reverse Chain Rule” or “U-Substitution Method”. We can use this method to find an integral value when it is set up in the special form. It means that the given integral is of the form: ∫ f (g (x)).g' (x).dx = f (u).du. May 22, 2019 · Watch on. U-substitution in definite integrals is just like substitution in indefinite integrals except that, since the variable is changed, the limits of integration must be changed as well. If you don’t change the limits of integration, then you’ll need to back-substitute for the original variable at the end. Something of the form 1/√ (a² - x²) is perfect for trig substitution using x = a · sin θ. That's the pattern. Sal's explanation using the right triangle shows why that pattern works, "a" is the hypotenuse, the x-side opposite θ is equal to a · sin θ, and the adjacent side √ (a² - x²) is equal to a · cos θ .The second integral is more difficult because the first integral is simply a \(u\)-substitution type. 7.3: Trigonometric Substitution. Simplify the expressions in exercises 1 - 5 by writing each one using a single trigonometric function. 1) \(4−4\sin^2θ\) 2) \(9\sec^2θ−9\) AnswerIn this video, we talk about the method of U-Substitution to solve integrals. For more help, visit www.symbolab.com Like us on Facebook: https://www.facebook...Use our trig substitution table, and substitute x = tan(u). ... (1 + x 2) dx = cos 2 (u) dx = du Easy to integrate: ∫1/(1 + x 2) dx = ∫du = u + c = arctan(x) + c . This entry was posted in Integration by substitution, More Challenging Problems on June 30, 2017 by mh225. ...A lecture video about substitution or u-substitution method in the process of integration or antiderivatives of algebraic functions. These are the fundamenta... Aug 9, 2023 · U-Substitution and Integration by Parts U-Substitution R The general formR of 0an integrand which requires U-Substitution is f(g(x))g (x)dx. This can be rewritten as f(u)du. A big hint to use U-Substitution is that there is a composition of functions and there is some relation between two functions involved by way of derivatives. ExampleR √ 1In basic U substitution, the goal is to identify an inner function, find its derivative, and substitute to simplify the integral.. 2. Trigonometric U Substitution: This type of U substitution is employed when dealing with integrals involving trigonometric functions. It often involves identifying a trigonometric expression within the integral and using a …The payment in lieu of dividends issue arises in conjunction with the short sale of stocks. Short selling is a trading strategy to sell shares a trader does not own, and buy them b...To understand integration by substitution, you can just use the chain rule in reverse: \begin{equation} \int f(g (x)) g'(x) dx = F (g (x)) + C, \end{equation} where $ F $ is an anti derivative of $ f $. To check this, just take the derivative of …Integral Calculus (2017 edition) 12 units · 88 skills. Unit 1 Definite integrals introduction. Unit 2 Riemann sums. Unit 3 Fundamental theorem of calculus. Unit 4 Indefinite integrals. Unit 5 Definite integral evaluation. Unit 6 Integration techniques. Unit 7 Area & arc length using calculus. Unit 8 Integration applications.U Substitution for Definite Integrals. In general, a definite integral is a good candidate for u substitution if the equation contains both a function and that function’s derivative. When evaluating definite integrals, figure out the indefinite integral first and then evaluate for the given limits of integration. Example problem: Evaluate: Here is a summary for the sine trig substitution. √a2 − b2x2 ⇒ x = a bsinθ, − π 2 ≤ θ ≤ π 2. There is one final case that we need to look at. The next integral will also contain something that we need to make sure we can deal with. Example 5 Evaluate the following integral. ∫ 1 60 x5 (36x2 + 1)3 2 dx. Show Solution.THIS SECTION IS CURRENTLY ON PROGRESS. \ (u\) substitution is a method where you can use a variable to simplify the function in the integral to become an easier function to integrate. This technique is actually the reverse of the chain rule for derivatives.Jan 6, 2021 ... "Double Substitution" is a term I coined myself, but that simply refers to problems where you have to solve for x in your "u=f(x)" statement&nbs...Substitution for Definite Integrals. Substitution can be used with definite integrals, too. However, using substitution to evaluate a definite integral requires a change to the limits of integration. If we change variables in the integrand, the limits of integration change as well. In calculus, the integration by substitution method is also known as the “Reverse Chain Rule” or “U-Substitution Method”. We can use this method to find an integral value when it is set up in the special form. It means that the given integral is of the form: ∫ f (g (x)).g' (x).dx = f (u).du. U Substitution for Definite Integrals. In general, a definite integral is a good candidate for u substitution if the equation contains both a function and that function’s derivative. When evaluating definite integrals, figure out the indefinite integral first and then evaluate for the given limits of integration. Example problem: Evaluate: u= sin x alternatively you may make t-formula substitution so you bring an expression to some algebraic form so you could split it up using partial fraction. There is also integration parts although in that case you would substitute u= G (x) so you can integrate f (x)g (x) using a formula similar to the product rule.Course: Class 12 math (India) > Unit 9. Lesson 6: u-substitution. 𝘶-substitution intro. 𝘶-substitution: rational function. 𝘶-substitution: multiplying by a constant. 𝘶-substitution: logarithmic function. 𝘶-substitution: challenging application. 𝘶-substitution warmup.The first piece will be something you can use u substitution for to get the integral into a form you know how to integrate. The second piece will be the derivative of whatever you …𝘶-substitution: definite integrals. 𝘶-substitution with definite integrals. 𝘶-substitution: definite integrals. 𝘶-substitution: definite integral of exponential function. Math > AP®︎/College Calculus AB > Integration and accumulation …Identifying which function to take as 'u' simply comes with experience. Some integrals like sin (x)cos (x)dx have an easy u-substitution (u = sin (x) or cos (x)) as the 'u' and the derivative are explicitly given. Some like 1/sqrt (x - 9) require a trigonometric ratio to be 'u'. Some other questions make you come up with a completely (seemingly ...
, Sal integrates the u-substitution in the usual fashion and it makes sense that he uses the boundaries x = 2 to x = 1 because the problem is a definite integral. I guess my question is if you integrated the u-substitution as an indefinite integral you would get (u^4)/4 + C but the C goes away when you've constricted it to a set of boundaries.. Feeling whitney lyrics
The Integral Calculator solves an indefinite integral of a function. You can also get a better visual and understanding of the function and area under the curve using our graphing tool. Integration by parts formula: ? u d v = u v-? v d u. Step 2: Click the blue arrow to submit. Choose "Evaluate the Integral" from the topic selector and click to ...Rewrite the integral (Equation 2.7.1) in terms of u: ∫(x2 − 3)3(2xdx) = ∫u3du. Using the power rule for integrals, we have. ∫u3du = u4 4 + C. Substitute the original expression for x back into the solution: u4 4 + C = (x2 − 3)4 4 + C. We can generalize the procedure in the following Problem-Solving Strategy.Learn ALL calculus 2 integral techniques u-substitution, trigonometric substitution, integration by parts, partial fraction decomposition, non elementary int...Integration by substitution - also known as the "change-of-variable rule" - is a technique used to find integrals of some slightly trickier functions than standard integrals. It is useful for working with functions that fall into the class of some function multiplied by its derivative.. Say we wish to find the integral. #int_1^3ln(x)/xdx#Soylent is coming to 7-Eleven. Food-hacking is coming to 7-Eleven. The convenience store chain is set to begin selling bottles of Soylent, the liquid meal replacement marketed to p...since you start with f'(g(x))*g'(x) you ust have to take the integral of f'(g(x)) to get f(g(x)), though it's easier if g(x) is just a single variable, so we substitute in u for g(x). of course at …Description. example. G = changeIntegrationVariable( F , old , new ) applies integration by substitution to the integrals in F , in which old is replaced by new ...Jan 12, 2024 · Solution. We'll need substitution to find an antiderivative, so we'll need to handle the limits of integration carefully. Let's solve this example both ways. Step One – find the antiderivative, using substitution: Let u = 3 x − 1. Then d u = 3 d x and. ∫ ( 3 x − 1) 4 d x = ∫ u 4 ( 1 3 d u) = 1 3 u 5 5 + C.In this video, we talk about the method of U-Substitution to solve integrals. For more help, visit www.symbolab.com Like us on Facebook: https://www.facebook...The method of “ [latex]u [/latex]-substitution” is a way of doing integral problems that undo the chain rule. It also helps deal with constants that crop up. Identify an “inside” function whose derivative is multiplied on the outside, possibly with a different constant. Call this “inside” function [latex]u [/latex].What a u-substitution does is that it creates a map from the x world to the u world (i.e. the substitution we make maps every value of x to a corresponding value of u). As a result, …, Sal integrates the u-substitution in the usual fashion and it makes sense that he uses the boundaries x = 2 to x = 1 because the problem is a definite integral. I guess my question is if you integrated the u-substitution as an indefinite integral you would get (u^4)/4 + C but the C goes away when you've constricted it to a set of boundaries.Identifying which function to take as 'u' simply comes with experience. Some integrals like sin (x)cos (x)dx have an easy u-substitution (u = sin (x) or cos (x)) as the 'u' and the …After the substitution, u is the variable of integration, not x. But the limits have not yet been put in terms of u, and this is essential. 4. (nothing to do) u = x ³−5. x = −1 gives u = −6; x = 1 gives u = −4. 5. The integrand still contains x (in the form x ³). Use the equation from step 1, u = x ³−5, and solve for x ³ = u +5.Integration by substitution. In calculus, integration by substitution, also known as u-substitution, reverse chain rule or change of variables, [1] is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation, and can loosely be thought of as using the chain rule "backwards." Using 𝘶-substitution in a situation that is a bit different than "classic" 𝘶-substitution. In this case, the substitution helps us take a hairy expression and ...Intuit QuickBooks recently announced that they introducing two new premium integrations for QuickBooks Online Advanced. Intuit QuickBooks recently announced that they introducing t...Course: AP®︎/College Calculus AB > Unit 6. Lesson 11: Integrating using substitution. 𝘶-substitution intro. 𝘶-substitution: multiplying by a constant. 𝘶-substitution: defining 𝘶. 𝘶-substitution: defining 𝘶 (more examples) 𝘶-substitution. 𝘶-substitution: definite integral of exponential function. Math >. The integration by substitution class 12th is one important topic which we will discuss in this article. In the integration by substitution,a given integer f (x) dx can be changed into another form by changing the independent variable x to z. This is done by substituting x = k (z). Consider I = f (x)dx. Now substitute x = k (z) so that dx/dz ....