Recursive version of integration by parts
General description of technique
The general procedure is as follows:
- Start doing the integration by parts.
- If necessary, apply integration by parts twice OR use some trigonometric identity with the ultimate goal of seeing the original integration appear again. Note that you should take care to avoid the circular trap.
- Denote by the particular choice of antiderivative for which the left and right sides are equal on the nose,i.e., not just up to additive constants.
- Solve the linear equation in .
- The general indefinite integral is this solution plus an arbitrary additive constant.
For further information, refer: Sine-squared function#Integration
There are many ways of integrating . One of these uses the recursive version of integration by parts. This method is given below:
We now rewrite and obtain:
Setting to be a choice of antiderivative so that the above holds without any freely floating constants, we get:
Rearranging, we get:
So the general antiderivative is:
For further information, refer: Secant-cubed function#Integration
We rewrite and perform integration by parts, taking as the part to integrate. We use that an antiderivative of is whereas the derivative of is :
We now use the fact that , or more explicitly, , to rewrite this as:
We now use the integration of the secant function to simplify this as:
We can choose an antiderivative of so that the above equality (between the left-most and right-most expression) holds without any additive constant adjustment, and we get:
We rearrange and obtain:
Dividing by 2, we get:
The general antiderivative expression is thus:
Exponential times cosine function
Consider the integration problem:
We proceed by taking as the part to integrate and as the part to differentiate. We get:
To avoid the circular trap, we should pick the exponential function as the part to integrate again, getting:
Now, we take as the antiderivative where the above holds without additive constants, and get:
Rearranging, we get:
This gives us:
So the general integral is: