Integration Using the U-Substitution

Introduction

Integration by substitution (or u-\text{substitution}) is most effective when an integral contains a composite function whose inner derivative also appears in the integrand. In such cases, setting u=g(x) transforms

allowing you to apply basic integration rules directly. Note that u-substitution is done to make sure that the integral is solved using the standard formulas, but with a different variable.

Illustrative Example #1

Evaluate the integral \displaystyle \int x^4 (x^5 + 3)^6 \, dx

First, choose the inner function and compute its differential:

Substitute into the original integral:

Since the integrand is now a simple power of u, apply the power rule:

Finally, substitute back u = x^5 + 3

Illustrative Example #2

Evaluate the integral \displaystyle \int \sqrt{\,9 + 5x\,}\,dx.

First, set the inner expression as the new variable and find its differential:

Rewrite the integrand in terms of u:

Now apply the power rule to integrate u^{\frac{1}{2}}

Finally, substitute back u = 9 + 5x:

Illustrative Example #3

Evaluate the integral \displaystyle \int \frac{1}{\,(2 - 5t)^{3}\,}\;dt.

First, let the inner expression be the new variable and compute its differential:

Rewrite the integrand in terms of u:

Now apply the power rule to integrate u^{-3}:

Finally, substitute back u = 2 - 5t:

Illustrative Example #4

Evaluate the integral \displaystyle \int e^x\,\cos\bigl(e^x\bigr)\,dx.

First, set the inner function equal to \(u\) and find its differential:

Substitute into the original integral:

Integrate \cos(u) with respect to u:

Finally, substitute back u = e^x:

Illustrative Example #5

Evaluate the integral \displaystyle \int x\,\sin\bigl(x^2\bigr)\,dx.

First, set the inner function as the new variable and compute its differential:

Substitute into the original integral:

Integrate \sin(u) with respect to u:

Finally, substitute back u = x^2:

Illustrative Example #6

Evaluate the integral \displaystyle \int \sec^{2}\bigl(\tfrac{1}{x}\bigr)\,\frac{1}{x^{2}}\,dx.

First, let the inner function be the new variable and compute its differential:

Substitute into the original integral:

Integrate \sec^{2}(u) withe respect to u:

Finally, substitute back u = \tfrac{1}{x}:

Illustrative Example #7

Evaluate the integral

First, set the inner function equal to \(u\) and compute its differential:

Substitute into the integral:

Integrate \csc(u)\,\cot(u) with respect to u:

so

Finally, substitute back u = 2\pi t:

Illustrative Example #8

Evaluate the integral \displaystyle \int x\,e^{-\,3x^{2}}\,dx.

First, choose the inner function and compute its differential:

Substitute into the integral:

Integrate e^{u}with respect to u:

Finally, substitute back u = -3x^{2}:

Illustrative Example #9

Evaluate the integral

Let the inner function be

Substitute into the integral:

Integrate \cos(u) with respect to u:

Finally, substitute back u = \sin(x):

Illustrative Example #10

Evaluate the integral

First, let the inner function be

Substitute into the integral:

Integrate u with respect to u:

Finally, substitute back u = \arctan(x):

Illustrative Example #11

Evaluate the integral

First, set the inner function equal to u and compute its differential:

Substitute into the integral:

Integrate \sin(u) with respect to u:

Finally, substitute back u = \ln(x):

Illustrative Example #12

Evaluate the integral

First, let the inner expression be the new variable and compute its differential:

Substitute into the integral:

Integrate 1/u with respect to u:

Finally, substitute back u = 1 + e^x: