Math Core

Integration by parts, partial fractions and improper integrals all end the same way: you reduce a hard integral to one you already know how to do. So before you learn the new BC techniques, make sure the AB toolkit is fast and automatic. This lesson reviews the basic antiderivatives, $u$ -substitution, and two algebra moves (long division and completing the square) that turn unfamiliar integrands into familiar ones.

The basic antiderivatives

Every technique in this unit eventually hands you one of these. You should be able to write each one without thinking.

Integrand	Antiderivative
$x^n$ , $n \ne -1$	$\dfrac{x^{n+1}}{n+1} + C$
$\dfrac{1}{x}$	$\ln\lvert x\rvert + C$
$e^x$	$e^x + C$
$\sin x$	$-\cos x + C$
$\cos x$	$\sin x + C$
$\sec^2 x$	$\tan x + C$
$\sec x \tan x$	$\sec x + C$
$\dfrac{1}{1 + x^2}$	$\arctan x + C$
$\dfrac{1}{\sqrt{1 - x^2}}$	$\arcsin x + C$

When the inside is a linear expression $ax + b$ instead of plain $x$ , divide by $a$ : for example, $\displaystyle\int \cos(3x)\,dx = \tfrac{1}{3}\sin(3x) + C$ . You can always check an antiderivative by differentiating it.

u-substitution

Substitution undoes the chain rule. Look for an "inside" function whose derivative also appears (up to a constant factor) as a multiplier.

u-substitution

If $u = g(x)$ , then $du = g'(x)\,dx$ and

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

For a definite integral, change the limits too: $\displaystyle\int_a^b f\big(g(x)\big)\,g'(x)\,dx = \int_{g(a)}^{g(b)} f(u)\,du$ .

Changing the limits means you never have to go back to $x$ . If you prefer to keep the $x$ -limits, you must rewrite the antiderivative in terms of $x$ before you plug in.

Worked example: A basic substitution

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

Let $u = x^2$ , so $du = 2x\,dx$ and $x\,dx = \tfrac{1}{2}\,du$ .

\int x\,e^{x^2}\,dx = \frac{1}{2}\int e^u\,du = \frac{1}{2}e^u + C = \frac{1}{2}e^{x^2} + C.

Check: $\dfrac{d}{dx}\left(\tfrac{1}{2}e^{x^2}\right) = \tfrac{1}{2}e^{x^2}\cdot 2x = x\,e^{x^2}$ .

Worked example: A definite integral with new limits

Evaluate $\displaystyle\int_0^1 \frac{x}{x^2 + 1}\,dx$ .

Let $u = x^2 + 1$ , so $du = 2x\,dx$ . When $x = 0$ , $u = 1$ ; when $x = 1$ , $u = 2$ .

\int_0^1 \frac{x}{x^2 + 1}\,dx = \frac{1}{2}\int_1^2 \frac{du}{u} = \frac{1}{2}\Big[\ln u\Big]_1^2 = \frac{1}{2}(\ln 2 - 0) = \frac{\ln 2}{2}.

Common mistake

When you substitute in a definite integral, either change the limits to $u$ -values or switch back to $x$ before evaluating. Plugging the original $x$ -limits into an antiderivative written in $u$ is the most common substitution error on the AP exam.

Long division for improper fractions

A rational function $\dfrac{p(x)}{q(x)}$ is improper when the degree of the numerator is at least the degree of the denominator. Divide first. The quotient is a polynomial (easy to integrate) and the remainder is a proper fraction.

Worked example: Divide, then integrate

Find $\displaystyle\int \frac{x^2 + 1}{x + 1}\,dx$ .

Divide: $x^2 + 1 = (x - 1)(x + 1) + 2$ , so

\frac{x^2 + 1}{x + 1} = x - 1 + \frac{2}{x + 1}.

Now integrate term by term:

\int \left(x - 1 + \frac{2}{x + 1}\right)dx = \frac{x^2}{2} - x + 2\ln\lvert x + 1\rvert + C.

You'll use this same first step in the partial fractions lesson whenever the fraction is improper.

Completing the square

A quadratic denominator that doesn't factor over the real numbers often leads to arctangent. Complete the square to get the form $u^2 + a^2$ , then use

\int \frac{du}{u^2 + a^2} = \frac{1}{a}\arctan\left(\frac{u}{a}\right) + C.

Worked example: An arctangent integral

Find $\displaystyle\int \frac{dx}{x^2 + 4x + 5}$ .

Complete the square: $x^2 + 4x + 5 = (x^2 + 4x + 4) + 1 = (x + 2)^2 + 1$ . With $u = x + 2$ and $du = dx$ ,

\int \frac{dx}{(x + 2)^2 + 1} = \int \frac{du}{u^2 + 1} = \arctan(x + 2) + C.

Choosing a method

Before you start any integral, take a few seconds to classify it:

Is it a basic form, possibly after simplifying or splitting a fraction? Integrate directly.
Is there an inside function with its derivative nearby? Use substitution.
Is it a rational function with numerator degree at least denominator degree? Divide.
Is the denominator an irreducible quadratic? Complete the square.
Is it a product of two different kinds of functions, such as $x e^x$ or $x \ln x$ ? That calls for integration by parts, the next lesson.
Is it a proper fraction with a denominator that factors into distinct linear factors? That calls for partial fractions, coming up after that.

Tip

On multiple-choice questions you can often skip the integration entirely: differentiate each answer choice and see which one gives back the integrand.

Practice

Practice 1

$\displaystyle\int \cos(3x)\,dx =$

Practice 2

Evaluate $\displaystyle\int_0^2 \left(3x^2 - 2x\right)dx$ .