Primitive function/Integration by parts/Section

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denote continuously differentiable functions. Then

Due to the product rule, the function is a primitive function for . Therefore,

In using integration by parts, two things are to be considered. Firstly, the function to be integrated is usually not in the form , but just as a product (if there is no product, then this rule will probably not help, however, sometimes the trivial product might help). Then for one factor, we have to find a primitive function, and we have to differentiate the other factor. If is a primitive function of , then the formula reads

Secondly, integration by parts only helps when the integral on the right, i.e. , can be integrated.

We determine a primitive function for the natural logarithm , with integration by parts. We write , and we integrate the constant function , and we differentiate the logarithm. Then

So a primitive function is .

A primitive function for the sine function is . In order to find a primitive function for , we use integration by parts to get a recursive relation to a power with a smaller exponent. To make this more precise, we work over an interval, the primitive function shall start at and have the value there. For , with integration by parts, we get

Multiplication with and rearranging yields

In particular, for , we have