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:

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

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