a lecture on integration by substitution


The Method of Substitution (or Change of Variables)

The following is a general method to find indefinite integrals that look like the result of a chain ruleMathworldPlanetmath.

  • When to use it: We use the method of substitution for indefinite integrals which look like the result of a chain rule. In particular, try to use this method when you see a compositionMathworldPlanetmathPlanetmath of two functions.

  • How to use it: In this method, we go from integrating with respect to x to integrating with respect to a new variable, u, which makes the integralDlmfPlanetmath much easier.

    1. (a)

      Find inside the integral the composition of two functions and set u= “the inner function”.

    2. (b)

      We also write du=dudxdx.

    3. (c)

      Substitute everything in the integral that depends on x in terms of u.

    4. (d)

      Integrate with respect to u.

    5. (e)

      Once we have the result of integration in terms of u (+C), substitute back in terms of x.

The method is best explained through examples:

Example 0.1.

We want to find e2x𝑑x. The integrand is e2x, which is a composition of two functions. The inner function is 2x so we set:

u=2x,du=2dx

Thus,

x=u/2,dx=du/2

Substitute into the integral:

e2x𝑑x=eudu2=12eu𝑑u=12eu+C=12e2x+C

The following are typical examples where we use the subsitution method:

Example 0.2.
xe3x2+7𝑑x

The inner function is u=3x2+7 and du=6xdx. Thus dx=du/(6x). Substitute:

xe3x2+7𝑑x=xeu6x𝑑u=eu6𝑑u=eu6+C=e3x2+76+C.
Example 0.3.
sin(3x+7)𝑑x

The inner function is u=3x+7 and du=3dx. Therefore:

sin(3x+7)𝑑x=sin(u)3𝑑u=-cos(u)3+C=-cos(3x+7)3+C.
Example 0.4.
(2x+3)x2+3x+20𝑑x

Inner u=x2+3x+20 and du=(2x+3)dx. Thus:

(2x+3)x2+3x+20𝑑x=u𝑑u=u1/2𝑑u=2u3/23+C=2(x2+3x+20)3/23+C.

Now another integral which is a little more difficult:

Example 0.5.
cos(lnx)x𝑑x

The inner function here is u=lnx and du=1xdx.

cos(lnx)x𝑑x=cos(u)1x𝑑x=cos(u)𝑑u=sin(u)+C=sin(lnx)+C.
Example 0.6.
3x2+14x+1x3+7x2+x+115𝑑x

This function is also a typical example of integration with substitution. Whenever there is a fraction, and the numerator looks like the derivative of the denominator, we set u to be the denominator:

u=x3+7x2+x+115,du=(3x2+14x+1)dx

Thus:

3x2+14x+1x3+7x2+x+115𝑑x=1u𝑑u=lnu+C=ln(x3+7x2+x+115)+C.
Example 0.7.
71+3x𝑑x

As in the example above, we set u=1+3x, du=3dx:

71+3x𝑑x=7udu3=731u𝑑u=73lnu+C=73ln(1+3x)+C.
Example 0.8.
t3(t4-50)700𝑑t

Here the inner function is u=t4-50 and du=4t3dt. Thus

t3(t4-50)700𝑑t=u7004𝑑u=14u701701+C=(t4-50)7014701+C.

Some other examples (solve them!):

exsin(ex)𝑑x,exex+1𝑑x,1xlnx𝑑x
Title a lecture on integration by substitution
Canonical name ALectureOnIntegrationBySubstitution
Date of creation 2013-03-22 15:38:29
Last modified on 2013-03-22 15:38:29
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 4
Author alozano (2414)
Entry type Feature
Classification msc 26A36
Related topic ALectureOnIntegrationByParts
Related topic ALectureOnTrigonometricIntegralsAndTrigonometricSubstitution
Related topic ALectureOnThePartialFractionDecompositionMethod