The reverse chain rule

Integrating functions of the form f'(x) times g(f(x)) by recognising a derivative, including powers, exponentials, logarithms, trigonometric functions and inverse trigonometric functions of a function.

The rule

$\displaystyle\int f'(x)\,[f(x)]^n\,dx$	$\dfrac{[f(x)]^{n+1}}{n + 1} + C$ (for $n \ne -1$)
$\displaystyle\int \frac{f'(x)}{f(x)}\,dx$	$\ln\|f(x)\| + C$
$\displaystyle\int f'(x)\,e^{f(x)}\,dx$	$e^{f(x)} + C$
$\displaystyle\int f'(x)\cos f(x)\,dx$	$\sin f(x) + C$
$\displaystyle\int \frac{f'(x)}{1 + [f(x)]^2}\,dx$	$\tan^{-1} f(x) + C$

Examples

Example 1●●

Find $\displaystyle\int 2x(x^2 + 1)^4\,dx$ and $\displaystyle\int x e^{x^2}\,dx$.

$2x$ is the derivative of $x^2 + 1$, so the first is $\dfrac{(x^2 + 1)^5}{5} + C$.

$x$ is half the derivative of $x^2$, so the second is $\tfrac12 e^{x^2} + C$.

Example 2●●

Find $\displaystyle\int \cos x\,\sin^3 x\,dx$ and $\displaystyle\int \frac{e^x}{1 + e^{2x}}\,dx$.

$\cos x$ is the derivative of $\sin x$, so the first is $\dfrac{\sin^4 x}{4} + C$.

$e^x$ is the derivative of $e^x$, and $e^{2x} = (e^x)^2$, so the second is $\tan^{-1}(e^x) + C$.

Flashcards

$displaystyleint f'(x)ig(f(x)ig)^n,dx$

$\dfrac{\big(f(x)\big)^{n+1}}{n + 1} + C$

$displaystyleint rac{f'(x)}{f(x)},dx$

$\ln|f(x)| + C$

$displaystyleint f'(x)e^{f(x)},dx$

$e^{f(x)} + C$

Practice

Evaluate $\displaystyle\int_0^1 2x(x^2 + 1)^3\,dx$.
Evaluate $\displaystyle\int_0^1 x e^{x^2}\,dx$. Type e for $e$.
Evaluate $\displaystyle\int_0^{\frac{\pi}{2}} \cos x\,\sin^2 x\,dx$.
Evaluate $\displaystyle\int_0^1 \frac{2x}{x^2 + 1}\,dx$. Type ln for $\ln$.
Evaluate $\displaystyle\int_0^{\frac{\pi}{2}} \sin x\cos x\,dx$.
Evaluate $\displaystyle\int_0^1 \frac{2x}{1 + x^4}\,dx$. Type pi for $\pi$.
$\displaystyle\int \frac{e^x}{1 + e^{2x}}\,dx = $
Which integral can be done with the reverse chain rule?
$\displaystyle\int \cos x\,e^{\sin x}\,dx = $