What is U substitution: Definition and 54 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. karush

    Solving this integral with u substitution

    Evaluate ##\displaystyle\int_{0}^{3}\frac{x+3}{\sqrt{x^{3}+1}}dx+5## W|A returned 11.7101 ok subst is probably just one way to solve this so ##u=x^{3}+1 \quad du= 3x^2##
  2. karush

    MHB 4.1.1 AP calculus Exam Int with U substitution

    Evaluate $\displaystyle\int{\dfrac{{(1-\ln{t})}^2}{t} dt=}$ $a\quad {-\dfrac{1}{3}{(1-\ln{t})}^3+C} \\$ $b\quad {\ln{t}-2\ln{t^2} +\ln{t^3} +C} \\$ $c\quad {-2(1-\ln{t})+C} \\$ $d\quad {\ln{t}-\ln{t^2}+\dfrac{(\ln{t^3})}{3}+C} \\$ $e\quad {-\dfrac{(1-\ln{t^3})}{3}+C}$ ok we can either expand...
  3. karush

    MHB 4.2.236 AP calculus Exam integral with u substitution

    AP Calculas Exam Problem$\textsf{Using $\displaystyle u=\sqrt{x}, \quad \int_1^4\dfrac{e^{\sqrt{x}}}{\sqrt{x}}\, dx$ is equal to which of the following}$ $$ (A)2\int_1^{16} e^u \, du\quad (B)2\int_1^{4} e^u \, du\quad (C) 2\int_1^{2} e^u \, du\quad (D) \dfrac{1}{2}\int_1^{2} e^u \, du\quad...
  4. Lazy Rat

    Integration problem using u substitution

    Homework Statement ## \int {sin} \frac{\pi x} {L} dx ##Homework Equations u substitution The Attempt at a Solution If i make ## u = \frac{\pi x} {L} ## and then derive u I get ## \frac {\pi}{L} ## yet the final solution has ## \frac {L}{\pi} ## The final solution is ## \frac {L}{\pi} - cos...
  5. Draconifors

    Triple integral using cylindrical coordinates

    Homework Statement The first part of the question was to describe E the region within the sphere ##x^2 + y^2 + z^2 = 16## and above the paraboloid ##z=\frac{1}{6} (x^2+y^2)## using the three different coordinate systems. For cartesian, I found ##4* \int_{0}^{\sqrt{12}} \int_{0}^{12-x^2}...
  6. karush

    MHB W.8.7.23 int trig u substitution

    $\tiny{Whitman \ 8.7.23}$ \begin{align} \displaystyle I&=\int \sin^3(t) \cos^2(t) \ d{t} \\ u&=\cos(t) \therefore du=-\sin(t) \, dt \\ \textit{substitute $\cos(t)=u$}&\\ I_u&=-\int (1-u^2) u^2 \, du=-\int(u^2-u^4) \, dt\\ \textit{integrate}&\\ &=-\left[\frac{u^3}{3}-\frac{u^5}{5}\right]...
  7. karush

    MHB S6-7.1.79 log integral u substitution

    $\large{7.R.79} $ $\tiny\text{UHW 242 log integral }$ https://www.physicsforums.com/attachments/5717 $$\begin{align} \displaystyle x& = \frac{1}{u} & {u}^{2 }du&={dx } \end{align} $$ $$I=\int_{0}^{\infty} \frac {\ln\left({\frac{1}{u}}\right)} {1+\frac{1}{{u}^{2 }}} {u}^{2} \,du \\ Stuck🐮...
  8. T

    Solve Indefinite Integral: U Substitution

    Homework Statement Im looking over the notes in my lecture and the prof wrote, \int_{0}^{2} \pi(4x^2-x^4)dx=\frac{64\pi}{15} Im wondering what's the indefinite integral of this equation. Homework Equations using u substitution The Attempt at a Solution \int \pi(4x^2-x^4)dx= \pi \int...
  9. thegreengineer

    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...
  10. T

    Integration using u substitution

    Homework Statement Evaluate the integral of (x+1)5^(x+1)^2 Homework EquationsThe Attempt at a Solution I set my u=(x+1) making du=1dx. This makes it u*5^u^2. I integrated the first u to be ((x+1)^2/2) however I don't know what to do with the 5^u^2
  11. TyroneTheDino

    Integrating Using a Substituation

    Homework Statement Use the substitution ##u=\frac{\pi} {2}-x## evaluate the integral ##\int_0^\frac {\pi}{2} \frac {\sin x}{\cos x + \sin x}dx##. Homework Equations [/B] ##\cos (\frac {\pi}{2}-x)=\sin x## The Attempt at a Solution [/B]I start by plugging "u" into the equation making the...
  12. thegreengineer

    Integral calculus: integral variable substitution confusion

    Recently I started seeing integral calculus and right now we are covering the topic of the antiderivative. At first sign it was not very difficult, until we started seeing integral variable substitution. The problem starts right here: Let's suppose that we have a function like this: \int...
  13. T

    Is this valid when doing u substitution for integration?

    So I'm doing length of an arc in my calculus 1 class. After plugging everything in the arc length formula. Now I have this complicated function to integrate. Square root of (16x^8+8x^4+1)/16x^4. I took the denominator out of my square root and got 4x^2. Now I take u=4x^2. Du/2x =dx...
  14. M

    Using u substitution for integrating.

    So I am pretty bad at u substitution. I don't really get how to replace values with du or u. Can you please give me tips on how to do u substitution well? Thanks.
  15. jdawg

    Solving U-Substitution Problem with 2x√(2x-3) in Calculus

    Homework Statement ∫2x√(2x-3) dx Homework Equations The Attempt at a Solution u=2x du=2 dx 1/2∫u√(u-3) du Am I on the right track with this? I'm not really sure what to do next.
  16. T

    MHB Solving u Substitution Problem: Tim's Question

    Hi, I'm working on a u substitution problem so that. u = 3-x so that du = (-1) dx , or (-1) du = dx . With these equations you just switch content from one side to the other with no problems? Thanks, Tim
  17. P

    U substitution. Why -1/x^2 is my du?

    Homework Statement ∫(1/x^(2))(3+1/x)^(3) Homework Equations U substitution is the way to go here The Attempt at a Solution My problem is that I can't figure my du and what is next. I know which one it is but I don't know the reason for it. u=3+1/x du= I chose ln|x| first but...
  18. Z

    Integration U substitution then square it I think.

    1. ∫ 1/(2√(x+3)+x) 2. Not sure if I'm beginging this correctly or not but I get stuck. 3. Let u= √x+3 then u2 = x+3 2udu=dx dx=2√[(x+3) Therefore: ∫1/u2-3 Not sure where to go from here?
  19. J

    Integration question, u substitution

    The problem statement Evaluate the indefinite integral ∫\frac{\sqrt{x}}{\sqrt{x}-3}dx The attempt at a solution My first thought was to substitute u for √(x)-3, but then du would equal \frac{1}{2\sqrt{x}}dx, and there's no multiple of du in the integrand. Next, I tried splitting up...
  20. 3

    U substitution and integration

    Homework Statement use the substitution u= x+y and v=y-2x to evaluate double integral from ∫1-0∫(1−x) -(0) of (√x+y) (y−2x)^2 dydx Homework Equations integration tables I am assuming The Attempt at a Solution i tried to integrate directly but none of my integration tables match...
  21. J

    Integrating cot^4 x (csc^4 x) dx Using Identities and U Substitution

    ∫(cot^4 x) (csc^4 x) dx Wolfram wants to use the reduction formula, but I'm meant to do this just using identities and u substitution. I was thinking something along the lines of: =∫cot^4 x (cot^2 x + 1)^2 dx =∫cot^8 x + 2cot^6 x + cot^4 x dx but I don't know where to go from there.
  22. L

    Integration by u substitution for inverse trig formulas

    Homework Statement You know the U substitution proofs for inverse trig functions that go like this: \int\frac{1}{a^{2}+x^{2}}dx \int\frac{a\frac{1}{a}}{a(1+\frac{x^2}{a^2})}dx let u = x/a du= dx/a ... \frac{1}{a}tan^{-1}(x/a)+cI have searched google and can't find any of these proofs for...
  23. Z

    U-Substitution for ∫3xdx/√(1-2x)

    Homework Statement ∫3xdx/√(1-2x) Homework Equations The Attempt at a Solution so i tried making u=3x which makes du=3dx but that substitution doesn't get rid of the x unde the square root. i tried u=1-2x and that gives du=-2dx and that doesn't get rid of the x on top. So I'm...
  24. L

    How Do You Apply U-Substitution to ∫sin(x^5)dx?

    Homework Statement Use Part 2 of the Fundamental Theorem of Calculus to find the derivative. \int_3^x sin(t^{5}) \, dt Homework Equations The Attempt at a Solution I know the general idea of what I'm supposed to do as far as evaluate the indefinate integral and then do a subtraction of the...
  25. R

    Is U Substitution the Key to Solving Tricky Integration Problems?

    Homework Statement The Attempt at a Solution if x2 = u - 1, and if x3 = x2 * x, then x3 should equal (u-1)x, not .5(u-1). I'm assuming that they got u.5 because (x2+1).5 = (u-1+1).5 which is the same as u^.5
  26. R

    U substitution and integration by parts

    I would think because of this The following problem: At this stage they should use integration by parts: However, maybe integration by parts is only useful when one of the parts is e^x ln or a trigonometric formula.
  27. S

    Integration, u substitution with limits

    Homework Statement find ∫x/√(x+1).dx with limits 1 & 0 using substitution x = u^2 -1 Homework Equations The Attempt at a Solution dx = du x = u^2 -1 u = √( x+1) sub limits of 1 & 0 into u. Hence new limits of √2 & 1 Therefore, ∫ u^2 -1/ u = ∫ u - 1/u =...
  28. S

    How do I Integrate this u substitution with limits.

    Homework Statement Find ∫e^x/ (1+e^2x). dx , with limits ln 2 & 0 given u= e^x Homework Equations The Attempt at a Solution u= e^x du/dx = e^x dx= du/e^x sub limits of ln2 & 0 → u Hence, limits 2 & 1 Therefore, ∫u* (1+e^2x)^-1* du/e^x = ∫ u/ (u + e^3x) = ∫...
  29. S

    Integration, u substitution with limits

    [b]1. Homework Statement Evaluate ∫ 3x /(3x+1)^2.dx , with limits 1 & 0 using the sustitution u = 3x+1 Homework Equations The Attempt at a Solution u= 3x+1 du/dx = 3 dx = du/3 Therefore, ∫ 3x*(u)^-2 * du/3 = ∫ x* (u)^-2 Since u = 3x +1 Therefore, x =...
  30. S

    Integration - u substitution problem (Integration by parts?)

    Homework Statement Find the integral of 3x* (2x-5)^6*dx, let u= 2x -5. Homework Equations Im not sure if i am meant to use integration by parts or not?? I was able to do previous questions of the topic just using u sub to get rid of the first x variable. The Attempt at a Solution...
  31. J

    U substitution with definite integral

    Homework Statement make a u- substitution and integrate from u(a) to u(b) Homework Equations ∫[0,1] √(t^5+2t) (5t^4+2) dt The Attempt at a Solution u= t^5+2t du= 5t^4+2 u(1)=2 u(0)=0 ∫[0,1] √(u) du (2/3)(u)^3/2+c l[0,2] (2/3)( 0^5+2(0))^3/2- (2/3)(2^5+2(2))^3/2 0+...
  32. J

    How Can U-Substitution Simplify Trigonometric Integrals?

    Homework Statement use substitution to evaluate the integral Homework Equations 1)∫ tan(4x+2)dx 2)∫3(sin x)^-2 dx The Attempt at a Solution 1) u= 4x+2 du= 4 (1/4)∫4 tan(4x+2) dx ∫(1/4)tan(4x+2)(4dx) ∫ (1/4) tanu du (1/4)ln ltan(u)l +c 2) u=sinx du= cosx or u=x du = 1 ?
  33. S

    Probability - u substitution to find gamma function.

    Homework Statement \int_0^∞ x^2exp(-x/2) dx Homework Equations The Attempt at a Solution Using u substitution: u = x/2 du = 1/2 dx \int_0^∞ 4u^2exp(-u) du*2 = 8 \Gamma(3) = 8*3! = 48 But the correct answer is 16 when I plug it in Wolfram's definite integral...
  34. J

    U Substitution: Solving Definite Integral of [(x^2sinx)/(1+x^6)]*dx

    The problem: The definite integral of [(x^2sinx)/(1+x^6)]*dx on the interval -∏/2 ≤ x ≤ ∏/2 I need help figuring out what the u should be for substitution. I've been trying to make the (1+x^6) my u, but I don't know if this is what I should be doing.
  35. D

    Integration, U substitution help

    Homework Statement Hi I am having a few problems with the below u substitution can anyone help, In particular what to do with the integral of the u substitution? Homework Equations \int2x2 square root of 1-x3 dx, u = 1-x3 Any pointers would be appreciated Thanks D
  36. 1

    Is It Necessary to Substitute u in Integrals?

    I received no credit, resulting in an 84 for a few integral problems. I had correct final answers for everything. When I confronted my professor about this, he said it was because I didn't actually put "u" and "du" into the integral. Is that really always necessary? Why actually put the u in...
  37. B

    What is the Purpose of U Substitution in Solving Integrals?

    Homework Statement the integral of 1/(1-y)dy Homework Equations The Attempt at a Solution ln|1-y|+C however I believe you use u substitution as 1-y=U? Why is this so?
  38. H

    Integrals using u substitution

    Homework Statement Using substitution, find the integral of 32x2/(2x+1)3 Homework Equations The Attempt at a Solution I initially tried plugging u in for 32x2 but that wouldn't work because it won't cancel out with the problem below it anyway. I'm pretty sure we are not expected...
  39. W

    Integral - u substitution with arctan

    Homework Statement integral of 1/(x^2 + z^2)^(3/2) dx, where z is a constant Homework Equations The Attempt at a Solution I set u = arctan(x/z) so du = z/(x^2 + z^2) dx but now I'm honestly stuck.
  40. M

    U substitution or substitution by parts?

    Homework Statement ∫〖e^√x/√x dx〗 would this be a u substitution or a substitution by parts? Homework Equations The Attempt at a Solution
  41. L

    Using u substitution, which of the following is equivalent to this integral?

    Homework Statement Using the u substutituion u = 2x + 1, ∫(2x + 1)1/2dx (when x goes from 0 to 2) is equivalent to? Answer: (1/2)*∫(u)1/2du (when x goes from 1 to 5) Homework Equations The Attempt at a Solution If u is 2x + 1, then du = 2dx. Thus, I get (1/2)*∫(u)1/2du...
  42. M

    My new U substitution approach? is this legal?

    Allow me to explain my new theory, The "Mancini conjecture." Ok...lets say I have an integral like (4-x^2)^(1/2) dx. and letting u = 4-x^2, we get du/dx = -2x, and if I took the second derivative of du/dx...i would get -2 this would be ideal, because I would then have du'' = -2 dx, or -1/2...
  43. S

    Integrating Difficult Equations: U Substitution

    Homework Statement As I was reviewing some of my previuosly learned calculus I came across somthing that I had either forgoten how to do or was never taught. How do you take the integral of somthing like this \int \sqrt{\frac{9}{4}x+1} I don't know how to start with this one. Do I use U...
  44. James889

    U substitution and partial integration

    Hey, I need to evaluate \int_{1}^{5}(6-2x)\sqrt{5-x}dx So. \tex{Let ~~u} = 6-2x~~ \tex{then}~~ du = 2 \frac{1}{2}~du = dx New limits: x = 5 \longrightarrow u = -4 x = 1 \longrightarrow u = 4 now, -\frac{1}{2}\int^{4}_{-4} u*\sqrt{5-x} and now for the partial integration. u \sqrt{5-x}~~ -...
  45. F

    Integrating v^2/v^2+4 - Exploring Arctan & u Substitution

    Homework Statement how do i integrate v^2 / v^2 + 4 Homework Equations i understand this has something to do with arctan but if i use u substitution to let v=(u/2) so (on the bottom) it becomes (1/4)(1+(v/2)^2) there's still a v^2 on the top which the u substitution does not...
  46. V

    Prove two integrals are the same using U substitution

    Homework Statement If a and b are positive numbers, show that \int_0^1 x^a*(1-x)^b\,dx = \int_0^1 x^b*(1-x)^a\,dx using only U substitution.Homework Equations Just U substitution and the given equation--I can't use multiplication rules or anything like that; otherwise it would be easy.The...
  47. T

    Trigonometric Integration and U Substitution

    Hi, We were going over trigonometric integration in Calculus II the other day. I got the basic idea, but get lost when we're doing the u-substitution. We had a problem like this: \int cos^3 (x) dx Then we did: \int (1 - sin^2 (x)) cos(x) dx Starting u-substitution: u =...
  48. D

    Integration using u substitution and arctan

    so I'm having problems with the coefficients in this problem. \int(10z+8/z^2-8z+41)dz i know that the main chunk is (a)ln|(z-4)^2+25|+(b)arctan((z-4)/5) a and b are 5 and 32/5 respectively the problem is i can't seem to split up the top so that the first portion is the derivitive...
  49. A

    Simple Integration using U Substitution

    Homework Statement Find the indefinite integral. The antiderivative or the integral of (x^2-1)/(x^2-1)^(1/2)dx Homework Equations The Attempt at a Solution Tried using (x^2-1)^(1/2) as u and udu for dx and I solved for x but I am still left with a 1 on top not sure how to...