Recursive version of integration by parts

General description of technique

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Examples

Sine-squared function

For further information, refer: Sine-squared function#Integration

There are many ways of integrating $\sin ^{2}$. One of these uses the recursive version of integration by parts. This method is given below:

$\int \sin ^{2}x\,dx=(\sin x)(-\cos x)-\int (\cos x)(-\cos x)\,dx=-\sin x\cos x+\int \cos ^{2}x\,dx$

We now rewrite $\cos ^{2}x=1-\sin ^{2}x$ and obtain:

$\int \sin ^{2}x\,dx=-\sin x\cos x+\int (1-\sin ^{2}x)\,dx$

Setting $I$ to be a choice of antiderivative so that the above holds without any freely floating constants, we get:

$\!I=-\sin x\cos x+x-I$

Rearranging, we get:

$\!2I=x-\sin x\cos x$

This gives:

$I={\frac {x-\sin x\cos x}{2}}$

So the general antiderivative is:

${\frac {x-\sin x\cos x}{2}}+C$

Secant-cubed function

For further information, refer: Secant-cubed function#Integration

We rewrite $\sec ^{3}x=\sec x\cdot \sec ^{2}x$ and perform integration by parts, taking $\sec ^{2}$ as the part to integrate. We use that an antiderivative of $\sec ^{2}x$ is $\tan x$ whereas the derivative of $\sec x$ is $\sec x\tan x$:

$\int \sec ^{3}x\,dx=\int \sec x(\sec ^{2}x)\,dx=\sec x\tan x-\int (\sec x\tan x)\tan x\,dx=\sec x\tan x-\int \sec x\tan ^{2}x\,dx$

We now use the fact that $\tan ^{2}x+1=\sec ^{2}x$, or more explicitly, $\tan ^{2}x=\sec ^{2}x-1$, to rewrite this as:

$\!\int \sec ^{3}x\,dx=\sec x\tan x-\int \sec x(\sec ^{2}x-1)\,dx=\sec x\tan xx-\int \sec ^{3}x\,dx+\int \sec x\,d$

We now use the integration of the secant function to simplify this as:

$\!\int \sec ^{3}x\,dx=\sec x\tan x-\int \sec ^{3}x\,dx+\ln |\sec x+\tan x|$

We can choose an antiderivative $I$ of $\sec ^{3}$ so that the above equality (between the left-most and right-most expression) holds without any additive constant adjustment, and we get:

$\!I=\sec x\tan x-I+\ln |\sec x+\tan x|$

We rearrange and obtain:

$2I=\sec x\tan x+\ln |\sec x+\tan x|$

Dividing by 2, we get:

$\!I={\frac {\sec x\tan x+\ln |\sec x+\tan x|}{2}}$

The general antiderivative expression is thus:

${\frac {\sec x\tan x+\ln |\sec x+\tan x|}{2}}+C$