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:

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

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

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

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 $\sec x\tan x$:

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

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:

$\int {\sec }^{3}xdx=\sec x\tan x-\int \sec x({\sec }^{2}x-1)dx=\sec x\tan xx-\int {\sec }^{3}xdx+\int \sec xd$

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

$\int {\sec }^{3}xdx=\sec x\tan x-\int {\sec }^{3}xdx+\ln |\sec x+\tan x|$

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:

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