"U-substitution": The first option for difficult integrals

Integration by substitution, also called "u-substitution" (because many people who do calculus use the letter u when doing it), is the first thing to try when doing integrals that can't be solved "by eye" as simple antiderivatives.

Example 1

Integrate   $\int \frac{x \, dx}{x^2}$

Solution: This is an integral we already know. If we simply divide the numerator into the denominator of the integrand, we get

$$\int \frac{x \, dx}{x^2} = \int \frac{dx}{x} = ln|x| + C$$

Pretty easy. But let's do it again by substitution as a way of figuring the method out.

The substitution is

$$Let \; u = x^2, \; then \; \frac{du}{dx} = 2x$$

Now we rearrange a bit:

Example 2

Integrate:   $\int \frac{x \, dx}{x^2 + 7}$

Solution: Here's a kind of integral you'll get used to recognizing as a good candidate for u-substitution. Notice that the power of x in the denominator is one greater than that of the numerator. Because we'll be taking a derivative to do the substitution, the power of what's in the denominator will drop by one to match that of the numerator, and that could work. We'll make this substitution:

Let $u = x^2 + 7,$ then $du = 2x\, dx$

Example 3

Integrate   $\int tan(x) \, dx$

Solution: The trick here is to express the tangent function as a ratio of sine and cosine functions:

$$\int tan(x) \, dx = \int \frac{sin(x)}{cos(x)} \, dx$$

Then we can make this substitution:

Let $u = cos(x),$ then $du = -sin(x) \, dx$

How do we know to make that substitution? Intuition mostly, but if you guess wrong and let u = sin(x), you'll run into a brick wall, but did it really hurt? Probably not. Just try the other way. You didn't hurt the problem; it doesn't have feelings. The resulting simplified integral is

Example 4

Integrate   $\int x^2 \, sin(x^3) \, dx$

Solution: This is a great example of an integrand that contains powers of the independent variable that are one apart, x² and x³. That's always a good candidate for u-substitution.

Let $u = x^3,$ then $du = 3 x^2 \, dx$

The simplified integral and its solution are

$$\frac{1}{3} \int sin(u) \, du = -\frac{1}{3} cos(u) + C$$

Example 5

Integrate   $\int \frac{ln(x)}{x} \, dx$

Solution: In this example, consider the relationship between the derivative of ln(x) and 1/x. It's a good clue about how to do the substitution:

Let $u = ln(x),$ then $du = \frac{1}{x} \, dx$

The resulting integral couldn't be simpler:

$$\int u \, du = \frac{1}{2} + C$$

Example 6

U-substitution with definite integrals: Changing limits

Consider the definite integral

$$\int_1^3 \frac{x^2 \, dx}{(2 - x^3)^2}$$

Solution: The difference in powers between the term containing x³ and the x² in the numerator suggests this substitution:

Let $u = 2 - x^3,$ then $du = -3x^2 \, dx$

Plugging in we get the simplified integral

$$-\frac{1}{3} \int \frac{du}{u^2} = \frac{1}{3u}$$

Re-substitution of u = 2 - x³ gives

Evaluating at the limits is always a little tedious; it gives

$$ \begin{align} &= \frac{1}{3(2 - 27)} - \frac{1}{3(2 - 1)} \\ \\ &= \frac{-1}{75} - \frac{1}{3} = -\frac{13}{38} \end{align}$$

Example 7

Integrate   $\int_0^1 (2x - 3)^3$

Solution: Here is the (by now, hopefully) obvious substitution:

Let $u = (2x - 3),$ then $du = 2 \, dx$

Now we can quickly change the limits of integration, too, and just stick with u from here on:

$$ \begin{align} x = 0 \; &\rightarrow \; u = 2(0) - 3 = -3 \\[5pt] x = 1 \; &\rightarrow \; u = 2(1) - 3 = -1 \end{align}$$

The simplified integral is

Practice problems

Solve the following indefinite and definite integrals.

Three video examples

Here are three different examples of integration by U-substitution, spanning simple functions, logs and trigonometric functions. The trickiest things about U-substitution are (1) figuring out what to substitute – that's mostly trial-and-error and developing an intuition, and (2) figuring out how to properly substitute for the part that has "dx" in it.

Minutes of your life: 4:48