# What is Integration by substitution: Definition and 84 Discussions

In calculus, integration by substitution, also known as u-substitution or change of variables, 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".

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1. ### B Integration by parts of inverse sine, a solved exercise, some doubts...

Hi, PF, here goes an easy integral, meant to be an example of integration by parts. Use integration by parts to evaluate ##\int \sin^{-1}x \, dx## Let ##U=\sin^{-1}x,\quad{dV=dx}## Then ##dU=dx/\sqrt{1-x^2},\quad{V=x}## ##=x\sin^{-1}x-\int \frac{x}{\sqrt{1-x^2} \, dx}## Let ##u=1-x^2##...
2. ### Understanding Integration by Substitution

Not sure how do I start from here, but do I let $$u = lnx$$ and substitute? Cheers
3. ### A Analytical Integration of a Difficult Function

Is it possible to integrate the following function analytically? ##\int_{0}^{\infty} \frac{\exp{-(\frac{A}{\tau}+B\tau+\frac{A}{\beta-\tau})}}{\sqrt{\tau(\beta-\tau)}}d\tau,## where ##A##, ##B## and ##\beta## are real numbers. What sort of coordinate transformation makes the integral bounded...
4. ### B Integration by Substitution Using Infinite Sums

I have seen the wikipedia's proof which can be found here: https://proofwiki.org/wiki/Integration_by_Substitution However sometimes, we have problems where you have a ##d(x)## times ## f(g(x))## times g prime of x where we use substitution and it works but the proof didn't prove this...
5. ### Integration by substitution question

Homework Statement Question: To solve the integral ##\int \frac{1}{\sqrt{x^2-4}} \,dx## on an interval ##I=(2,+\infty)##, can we use the substitution ##x=\operatorname {arcsint}##? Explain Homework Equations 3. The Attempt at a Solution [/B] This is my reasoning, the function ##\operatorname...
6. ### I Multi-dimensional Integral by Change of Variables

Hi All, $$\int{\exp((x_2-x_1)^2+k_1x_1+k_2x_2)dx_1dx_2}$$ I can perform the integration of the integral above easily by changing the variable $$u=x_2+x_1\\ v=x_2-x_1$$ Of course first computing the Jacobian, and integrating over ##u## and ##v## I am wondering how you perform the change of...
7. ### Integrating with Substitution: Solving for ∫(3x^2+x)(2x^3+x^2)^2 dx

Homework Statement dz/dx=(3x2+x)(2x^3+x^2)^2[/B]Homework Equations ∫(3x^2+x)(2x^3+x^2)^2 dx The Attempt at a Solution I tried substituting (2x^3+x^2) Let t= 2x^3 + x^2 dt=6x^2+2x dx dt/dx= 6x^2+2x I can only solve till this point . I don't have any clue how to solve it further But how do we...
8. ### What substitution to use in this type of integral?

Homework Statement Find the integral \int \frac{3x+1}{(x^2-x-6)\sqrt{3x^2+4x-7}}\mathrm dx 2. The attempt at a solution I have tried the types of substitutions of irrational functions, and Euler substitutions. However, it seems that nothing simplifies this integral. What substitution is...
9. ### Integration of irrational function

Homework Statement Find the integral \int \frac{1}{(x-2)^3\sqrt{3x^2-8x+5}}\mathrm dx 2. The attempt at a solution I can't find a useful substitution to solve this integral. I tried x-2=\frac{1}{u},x=\frac{1}{u}+2,dx=-\frac{1}{u^2}du that gives \int \frac{1}{(x-2)^3\sqrt{3x^2-8x+5}}\mathrm...
10. ### Integration with power substitution

Homework Statement Find the integral \int \frac{2\sqrt[5]{2x-3}-1}{(2x-3)\sqrt[5]{2x-3}+\sqrt[5]{2x-3}}\mathrm dx 2. The attempt at a solution...
11. ### Change of boundaries for an integration by substitution....

Homework Statement Let: ##I=\int _{-1} ^{1}{\frac{dx}{\sqrt{1+x}+\sqrt{1-x}+2}}## Show that ##I=\int_{0}^{\frac{ \pi}{8}}{\frac{2cos4t}{cos^{2}t}}## using ##x=sin4t##. Hence show that ##I=2\sqrt{2}-1- \pi## Homework EquationsThe Attempt at a Solution The substitution is ##x=sin4t## which...
12. ### Is There a Hidden Meaning in Integrating the Lorentz Factor with Trig Sub?

Just for my own entertainment I integrated the Lorentz factor with respect to velocity, using basic trig sub, I got the equation arcsin(v/c)*(mc^2). What does this mean? Is it just useless and irrelevant in the physics world or does it have some sort of hidden meaning?
13. ### Integral of (1-x^2)/(1+x^2)*(1/(1+x^4)^(1/2))dx

Hi, I'm trying to solve ∫(x2-1)/(1+x2)*1/(1+x4)(1/2)dx I'm apparently meant to get some non-complex result, the question suggests to use the substitution u2 = x2 + 1/x2 But I haven't gotten anywhere with this. Any methods or suggestions (or the solution) would be much appreciated! Thanks
14. ### Substitution method for finding an integral's interval changes

Look, I was wondering if substituting the variable more than once is valid and hence the definite integral intervals change this way. Consider the following integral (I'm working for finding the volume of a solid of revolution): *\pi \int_{-3}^{5}3^{2}-(\sqrt{\frac{y+3}{2}}+1)^2dy Personally I...
15. ### Trig Integration By Substitution

Mod note: Moved from technical math section ∫(2x+6)/sqrt(5-4x-x^2) I have 2/3(ln|tan(theta)+sec(theta)|-3|cos(theta)|) where x=sin^-1((x+2)/3)
16. ### Integration by substitution for sin/cos products

Ok so I might be doing something silly but I just don't understand what is going on here. So the integral: i = ∫ sin x (cos x)^3 dx First I say u = cos x. So du = - sin x dx. So now I have i = ∫ - u^3 du. Which gives: i = -(1/4)u^4 or -(1/4)(cos x)^4. Easy. But if I say u = sin x...
17. ### How do I do this integration by substitution?

Homework Statement ∫1/(3+((2x)^.5))dx the answer should be ((2x)^.5) - 3ln(3+((2x)^.5)) + c I keep getting ((2x)^.5) - ln(3+((2x)^.5)) + c Homework Equations ∫1/(3+((2x)^.5))dx The Attempt at a Solution I did: u = 3 + ((2x)^.5) du = 1/((2x)^.5) dx du((2x)^.5) = dx...
18. ### Integration by substitution: Can I treat this as constant

I am trying to compute the following integral: \int \exp^{w^T \Lambda w}\, d\theta where \Lambda is a constant wrt \theta w = y - t(x, \theta) So, I am trying to use substitution and I have: d\theta = \frac{-dw}{t^{'}(x, \theta)} So, substituting it, I have the following integral...
19. ### Integration by substitution question

Homework Statement I've been working on a problem from Apostol "Calculus" Volume 1 (not homework but self study). The problem is Section 5.8, Number 25 (Page 217) and states: If [tex]$\m$[\tex] is a positive integer, show that: \int_0^{\frac{\pi}{2}} cos^m x sin^m x dx =...
20. ### Integration By Substitution Problem (Trig)

Homework Statement Integrate the following using substitution techniques ∫e3tcsc(e3t)cot(e3t) dt Homework Equations csc(t) = 1/sin(t) cot(t) = 1/tan(t) cot(t) = cos(t)/sin(t) 1 + cot2(t) = csc2(t) The Attempt at a Solution ∫e3tcsc(e3t)cot(e3t) dt set u = cot(e3t)...
21. ### Integration by substitution u=tan(t)

Homework Statement Integrate -1/(1+x(sin(t))^2) between 0 and pi/2 using the substitution u = tan(t)The Attempt at a Solution du/dt = (sec(t))^2 dt/du = 1/(1+u^2) I've messed around with the integral and trig. identities but I don't seem to be getting anywhere changing the integral to make...
22. ### Deriving the Formula for Integration by Parts

integration by parts I'm working through Apostol's Calculus. I have attached the problem. I need to derive the formula integrating by parts. It is not a hard problem, but I can't seem to understand how on Earth the author came up with that expression. I take f(x) = (a^2 - x^2)^n, so...
23. ### Integration by substitution diff. eq.

Homework Statement From Larson, 9th Edition: Section 4.5. Solve the differential equation \frac{\operatorname{d}y}{\operatorname{d}x}=4x+ \frac{4x}{\sqrt{16-x^2}}Homework Equations The Attempt at a Solution Well, I can get my book's answer, but not through doing things in the prescribed way...
24. ### Integration by substitution - I'm stuck

θHomework Statement I'm trying to do an integration by substitution, but I'm completely stuck at the moment ∫(1-sin2θ)cosθ dθ Homework Equations ∫u dv = uv - ∫v du The Attempt at a Solution u = 1 - sin2θ dv = cosθ dθ du = -2sinθcosθ or -sin(2θ) v = sin I found du as...
25. ### Need help solving this indefinite integral via integration by substitution

Homework Statement Calculate the following integral: \int{\frac{\sqrt{x+1}}{x+5}dx} \ , x ≥ 1 By using the following substitution: t=\sqrt{x+1} Homework Equations Well using the integration by substitution formula. The Attempt at a Solution So I have t=\sqrt{x+1}...
26. ### Integration by substitution problem

Homework Statement Using a suitable substitution find the solution to: ∫ (x+2)50(x+1)dx Homework Equations The Attempt at a Solution I can't find a solution to this using substitution. Wolfram alpha give an answer that is too long to be calculated by hand. Can anyone work...
27. ### Integration by substitution for dy/dx=(x+2y)/(3y-2x)

Homework Statement Original problem is differential equation dy/dx=(x+2y)/(3y-2x) This is part of solving differential equation. x(dv/dx) = (1+4v-3v^2)/(3v-2) so one way of solving, I take out the negative sign x(dv/dx) = -((3v^2-4v-1)/(3v-2)) , separate and bring over...
28. ### Integration by substitution (and esp. Weierstrass' substitution)

The stupid question of the day. Is it fair to say that\frac{du}{dx} = \frac 1 {dx/du} since this comes (I think) from the chain rule, \frac{dx}{du} \frac{du}{dx} = \frac{dx}{dx} = 1 Which means that, when integrating by substitution, I can choose to do either of \int f(u) du = \int...
29. ### Integration by substitution of sqrt cos theta.sin cube theta

Homework Statement To show that \int_{0}^ \frac{\pi}{2}\sqrt{cos\theta}sin^3(\theta) d\theta = 8/21 The Attempt at a Solution The above expression was simplified as \int_{0}^ \frac{\pi}{2}\sqrt{cos\theta}sin^2(\theta) sin(\theta) d\theta \int_{0}^...
30. ### Apostol's Integration by substitution problem

Homework Statement a) Show that: \int_{0}^{\pi} xf(sin (x))dx = \frac{\pi}{2}\int_{0}^{\pi} f(sin (x))dx [Hint: u = π - x] b) Use part a) to deduce the formula: \int_{0}^{\pi} \frac{xsin(x)}{1 + cos^2 (x)} dx = \pi\int_{0}^{1} \frac{dx}{1 + x^2} Homework Equations \int_{a}^{b}...
31. ### Where Did I Go Wrong? Evaluating an Integral Using Substitution

Hi, I'm new to integration and I'm trying to figure out where I went wrong on this question. I'm close to the answer, but I can't tell where I've gone wrong? Can anyone help? Thanks. Homework Statement Q. Evaluate the following: The Attempt at a Solution Please see attachment.
32. ### Assistance with Integration by substitution

Homework Statement I have the integral \int sin(2.13\sqrt{x}+2.4)\,dx I'm supposed to use the substitution y=2.13\sqrt{x}+2.4, aka. \sqrt{x}=\frac {y-2.4}{2.13} to gain the following description of the integral: \int sin(2.13\sqrt{x}+2.4)\,dx = E cos(y) + F\int y sin(y)\,dy I have...
33. ### Can substitution be used to find the indefinite integral of 2x/(x+5)^6?

Homework Statement Find the indefinite integral by substitution. ∫2x/(x+5)^6 dx Homework Equations The Attempt at a Solution I know how to do this using the method of partial fractions, but the book says to use substitution. Is there a way to just do a basic u-substitution...
34. ### A simple question about integration by substitution

Hello all, We've just begun integration in my maths class and I have a question about a certain aspect of integration by substitution. Let's say for instance you let u = 2x-1. Then you differentiate it and get du/dx = 2. My maths teacher said " you can now think of it as multiplying...
35. ### Definite Integration by Substitution

Homework Statement \int^2_1 6x\sqrt{x-1}dxHomework Equations The Attempt at a Solution Let u=x-1. Then, u+1=x, and du=dx. Continued from problem statement, =6 \int^1_0 (u+1)u^{\frac{1}{2}}du =6 \int^1_0 u^{\frac{3}{2}} + u^{\frac{1}{2}}du =6(1^{\frac{3}{2}} + 1^{\frac{1}{2}}) =6(2) =12 My Web...
36. ### Can x = a*sinh(t) be used for solving \int (1/((\sqrt{(x^2)+(a^2)}))^(3/2)*dx)?

Hi , I solved \int (1/((\sqrt{(x^2)+(a^2)}))^(3/2)*dx) using the substitution x = a*tan(\varphi I wonder if there are other methods to solve this problem? * (2/3) is the power on the radical function
37. ### So, for your problem, the correct value of f(-1) is 1/e, not e.

Homework Statement (i) find \int^{X}_{0} xe^{-x^{2}} dx in terms of X. (ii) Find \int^{X}_{0} xe^{-x^{2}} dx for X= 1, 2, 3 and 4. Homework Equations - The Attempt at a Solution (i) \int^{X}_{0} xe^{-x^{2}}dx -x^{2} = X dX/dx=-2x hence -1/2 dX = xdx so, -\frac{1}{2}...
38. ### Integration by substitution (I think)

Homework Statement Integral of d.cos j with regard to d.sin j Where d is a constant. Homework Equations The Attempt at a Solution I don't know how to approach this. I can substitute u=d.sin j Then I have Integral of dz/dj with regard to dz, but not sure where to go from here. Any help...
39. ### Substitution Rule for Integrals: Solving for the Unknown Variable

Hello! My problem is the following: Is \int_a^b f(z) dt = \int_{g(a)}^{g(b)} f(z) \frac{1}{g} dz ? \frac{dz}{dt} = g Thank you!
40. ### Trouble with Integrating \cos^{5}7x\sin7x Using Substitution

trying to solve the following integral by substitution but having trouble: \int\cos^{5}7x\sin7xdx I attempted to set u=\cos^{5}7x and ended up with (by chain rule...which I hope is correct!): du=-35\cos^{4}7x\sin7xdx This doesn't seem too helpful but can't think of a better...
41. ### Question about integration by substitution?

I am confused about integration in other cases, I understand that you can use substitution if the derivative exists next to what your trying to integrate then you can use it. However while studying Arc Length and surface of a revolution I came across a problem such that I had to integrate the...
42. ### Integration by substitution indefinite integral 5

Homework Statement indefinite integral 5\picos\pit Homework Equations The Attempt at a Solution 5\pi int cos\pit Substitution Method 5\pi x sin (1/\pit
43. ### Choosing the Right u: A Guide to Integration by Substitution

i don;t have a specific homework question. i have a sort of conceptual question instead when integrating by substitution, how do i know what to choose as u? for example integral of z^2 / (1 + z^3)^(1/3) dz i am suppposed to choose u as 1+z^3. any other value for u won't give me the...
44. ### Understanding Integration by Substitution

Homework Statement [PLAIN]http://img293.imageshack.us/img293/5026/solutoni.png Hi all, Can anyone explain what is going on where? I understand that it is a different way of writing the conventional integration by substitution, instead of using the symbol u. The second line, however...
45. ### Integration by substitution

Homework Statement evaluate: higher limit of 36 lower limit of 0 (36+3x)^1/2 dx Homework Equations i thought of using subsititution? The Attempt at a Solution g(x)=36+3x g'(x)=3 when x=0, u=36+3(0)=36 when x=36, u=36+3(36)=144 from lower limit of 36 to higher...
46. ### Integration by substitution where square root is U^2

Hi there, I am having difficulty with one aspect of intergration by substitution where the substituion of a square root is U^2, wondering if anyone can help. Problem: Integral of: 2x√(3x-4) dx by substituting U^2 = 3x-4 Would du^2/dx = 3 therefore 1/3 du^2 = dx (I think...
47. ### What is the Correct Integration by Substitution for \int \frac{3x}{2x+3}?

Homework Statement \int \frac{3x}{2x+3} u = 2x +3 x = \frac{1}{2}(u-3} ) dx = \frac{1}{2} du so now the integral should be, \int \frac{ \frac{3u-9}{2}}{u} \times \frac{1}{2} du = \frac{1}{2} \int \frac{3u-9}{2} \times \frac{1}{u} du \frac{1}{2} \int...
48. ### Solving Integrals Using Substitution: A Step-by-Step Guide

I'm attempting to solve the following problem: \int_{0}^{\infty} {\frac{x arctan(x)}{(1+x^{2})^{2}}dx} I started with a substitution: u=arctan(x), du=\frac{1}{(1+x^{2})}dx This seemed like the right thing to do, but after trying to put it together in several different ways I got...
49. ### Problem with Integration by substitution

Homework Statement If f is continuous and \int^{9}_{0}f(x)dx = 4, find \int^{3}_{0}xf(x^{2})dx Homework Equations None required The Attempt at a Solution Don't really know where to begin, but I tried: for \int^{3}_{0}xf(x^{2})dx let: u = x^{2} du = 2xdx substitute...
50. ### U-Substitution for Indefinite Integrals

Hi, am I on the right track with this U-substitution problem? Homework Statement Evaluate the indefinite integral Homework Equations integral of x^2(x^3 + 5)^9 dx The Attempt at a Solution integral of x^2(x^3 + 5)^9 dx Let u = x^3 + 5 du = 2x^2 1/2du = x^2 1/2 integral u^9 du 1/2...