Recognizing substitution forms

With a little practice, some forms scream "substitution". Two of them come up again and again and are worth knowing by heart.

The form $\dfrac{u'}{u}$. When the integrand is a fraction whose numerator is the derivative of the denominator, the antiderivative is a logarithm:

$$\int \frac{u'(x)}{u(x)}\,\mathrm{d}x = \ln|u(x)| + C$$

For example $\displaystyle\int \frac{2x}{x^2+1}\,\mathrm{d}x = \ln\left(x^2+1\right) + C$, because the top, $2x$, is the derivative of the bottom, $x^2+1$. This is the "logarithmic" antiderivative, to be spotted at a glance.

The form $u'u^n$. When a power of $u$ is multiplied by the derivative $u'$, we raise the power:

$$\int u'(x)\,u(x)^n\,\mathrm{d}x = \frac{u(x)^{n+1}}{n+1} + C \qquad (n \neq -1)$$

For example $\displaystyle\int 2x\left(x^2+1\right)^3\,\mathrm{d}x = \frac{\left(x^2+1\right)^4}{4} + C$.

Recognized form	Antiderivative
$\dfrac{u'}{u}$	$\ln\lvert u \rvert$
$u'u^n$, with $n \neq -1$	$\dfrac{u^{n+1}}{n+1}$
$u'e^u$	$e^u$
$u'\cos u$	$\sin u$

The common thread is always the same: look in the integrand for an inner function $u$ accompanied by its derivative $u'$. If you find it, substitution makes it melt away. If not, another technique will be needed — integration by parts, decomposition… Knowing how to recognize the form is half the work.