The antiderivatives to know by heart for the exam

The table

An antiderivative of $f$ is a function $F$ such that $F' = f$. Up to an additive constant.

$f(x)$	Antiderivative $F(x)$
$k$	$kx + C$
$x^n$ ($n \ne -1$)	$\dfrac{x^{n+1}}{n+1} + C$
$\dfrac{1}{x}$	$\ln
$e^x$	$e^x + C$
$\cos(x)$	$\sin(x) + C$
$\sin(x)$	$-\cos(x) + C$
$\dfrac{1}{\sqrt{x}}$	$2\sqrt{x} + C$

Never forget the + C

Every antiderivative is defined up to a constant. In an exercise you can often pick $C = 0$, but you must mention it.

The inverse move: powers

To integrate $x^n$, raise the exponent by 1, then divide by the new exponent.

$\int x^3 \, dx = \dfrac{x^4}{4} + C$.

Linear combinations

$\displaystyle \int (2x^3 + 5x - 1) \, dx = \dfrac{2x^4}{4} + \dfrac{5x^2}{2} - x + C = \dfrac{x^4}{2} + \dfrac{5x^2}{2} - x + C$.

The antiderivative of $\dfrac{u'}{u}$

It’s $\ln|u|$. Example: $\int \dfrac{2x}{x^2 + 1} \, dx = \ln(x^2 + 1) + C$ (the denominator stays positive here).

?Your turn

An antiderivative of f(x) = 3x² + 4x is:

A.6x + 4B.x³ + 2x²C.x³ + 4x²D.3x³ + 4x²

Check