Topic 1 of 7

Substitution and integration by parts

Step 16 · Calculus 2 topics

Undo the chain rule with substitution and the product rule with integration by parts, and learn to spot which one an integral needs.

Substitution

Substitution undoes the chain rule. If the integrand contains a function g(x) and also its derivative g′(x), set u = g(x). Then du = g′(x) dx, and the integral becomes simpler in u.

\int f(g(x))\,g'(x)\,dx = \int f(u)\,du

Example: A power

Find $\int 2x(x^{2} + 1)^{5}\,dx$ .

Let u = x² + 1, so du = 2x dx.
The integral becomes $\int u^{5}\,du = \frac{u^{6}}{6} + C$ .

Answer $\frac{(x^{2} + 1)^{6}}{6} + C$

Example: Fixing a constant

Find $\int x\,e^{x^{2}}\,dx$ .

Let u = x², so du = 2x dx and x dx is $\frac{1}{2}$ du.
$\frac{1}{2}\int e^{u}\,du = \frac{1}{2}e^{u} + C$ .

Answer $\frac{1}{2}e^{x^{2}} + C$

Example: A definite integral

Find $\int_{0}^{2} \frac{2x}{x^{2} + 1}\,dx$ .

Let u = x² + 1. When x = 0, u = 1; when x = 2, u = 5.
$\int_{1}^{5} \frac{du}{u} = \ln 5 - \ln 1$ .

Answerln 5, about 1.61

Integration by parts

Integration by parts undoes the product rule. Choose u to be a factor that gets simpler when you differentiate it, and dv to be the rest.

\int u\,dv = uv - \int v\,du

Example: x times eˣ

Find $\int x e^{x}\,dx$ .

Take u = x and dv = eˣ dx, so du = dx and v = eˣ.
$xe^{x} - \int e^{x}\,dx$ .

Answer $xe^{x} - e^{x} + C$

Example: A logarithm

Find $\int \ln x\,dx$ .

Take u = ln x and dv = dx, so du = dx ÷ x and v = x.
$x\ln x - \int x \cdot \frac{1}{x}\,dx$ .

Answer $x\ln x - x + C$

Example: Twice

Find $\int x^{2}\sin x\,dx$ .

Take u = x² and dv = sin x dx: $-x^{2}\cos x + \int 2x\cos x\,dx$ .
Parts again, with u = 2x: $\int 2x\cos x\,dx = 2x\sin x + 2\cos x$ .

Answer $-x^{2}\cos x + 2x\sin x + 2\cos x + C$

Choosing a method

Before integrating, look at the shape of the integrand.

You see	Try
A function and its derivative	Substitution
A power of x times eˣ, sin x or cos x	Parts, with u the power
ln x or an inverse trig function alone	Parts, with dv = dx
A form from the basic table	Integrate directly

Example: Spot the derivative

Which method suits $\int \frac{\ln x}{x}\,dx$ ?

$\frac{1}{x}$ is the derivative of ln x.
So u = ln x gives $\int u\,du = \frac{u^{2}}{2}$ .

Answer $\frac{(\ln x)^{2}}{2} + C$

Example: Check by differentiating

Check that $xe^{x} - e^{x}$ is an antiderivative of $xe^{x}$ .

By the product rule its derivative is eˣ + xeˣ − eˣ.
That leaves xeˣ.

AnswerIt checks

Try it yourself

Work each one on paper first, then open the answer to check.

Find $\int 3x^{2}(x^{3} - 4)^{4}\,dx$ .
Show answer
$\frac{(x^{3} - 4)^{5}}{5} + C$
Find $\int \cos(5x)\,dx$ .
Show answer
$\frac{1}{5}\sin(5x) + C$
Find $\int_{0}^{1} 2x\,e^{x^{2}}\,dx$ .
Show answer
e − 1, about 1.72
Find $\int x\cos x\,dx$ .
Show answer
$x\sin x + \cos x + C$
Find $\int_{1}^{e} \ln x\,dx$ .
Show answer
1
Which method suits $\int x^{3}e^{x}\,dx$ ?
Show answer
Integration by parts, three times, with u the power of x