Reverse chain rule

In a nutshell

The reverse chain rule makes use of the chain rule for differentiation to integrate functions that are of particular forms, such has $k [f(x)]^n f'(x)$ and $k\dfrac{f'(x)}{f(x)}$ .

Functions of the form $k\dfrac{f'(x)}{f(x)}$

When integrating functions of the form $k\dfrac{f'(x)}{f(x)}$ , try differentiating $y=\ln(f(x))$ and adjust for any constants.

Example 1

Evaluate $\int \dfrac{4x+2}{x^2+x+5}\, dx$ .

Try to differentiate the denominator to see if it matches the numerator:

$\dfrac{d}{dx}(x^2+x+5)=2x+1$

Note that this is very similar to the numerator - it differs by a constant factor. Therefore, try differentiating $y=\ln(x^2+x+2)$ and adjust for any constants:

$\dfrac{d}{dx}\ln(x^2+x+5)=\dfrac{2x+1}{x^2+x+5}$

$\Rightarrow \int \dfrac{2x+1}{x^2+x+5}\, dx = \ln(x^2+x+5)+C$

Use the fact that $4x+2=2\times(2x+1)$ :

$\begin{aligned} \int \dfrac{4x+2}{x^2+x+5}\, dx &= \int 2\times \dfrac{2x+1}{x^2+x+5}\, dx\\&= 2\times \int \dfrac{2x+1}{x^2+x+5}\, dx\\&=2\times \ln(x^2+x+5)+C\end{aligned}$

$\int \dfrac{4x+2}{x^2+x+5}\,dx = \underline{2 \ln(x^2+x+5)+C}$

Functions of the form $k [f(x)]^n f'(x)$

For functions of the form $k [f(x)]^n f'(x)$ , try differentiating $y=[f(x)]^{n+1}$ and adjust for any constants.

Example 2

Evaluate $\int \dfrac12\sin(x)\cos^3(x)\,dx$ .

Try to differentiate the function inside the power to see if it matches the other function:

$\dfrac{d}{dx}(\cos(x))=-\sin(x)$

This differs from the numerator by only a constant factor. Therefore, try differentiating $y=[\cos(x)]^4$ :

$\dfrac{d}{dx} (\cos^4(x))=-4\cos^3(x)\sin(x)$

$\Rightarrow \int -4\cos^3(x)\sin(x) \, dx= \cos^4(x)+C$

Adjust for the constant factor:

$\begin{aligned} \int \dfrac12\sin(x)\cos^3(x)\,dx &= -\dfrac{1}{8}\times \int -4\sin(x)\cos^3(x)\,dx\\&=-\dfrac18 \times \cos^4(x)+C \end{aligned}$

$\int \dfrac12\sin(x)\cos^3(x)\,dx =\underline{-\dfrac18 \cos^4(x)+C}$