linguistic:paraphrasing
Finding antiderivatives

Integration by parts is a heuristic rather than a purely mechanical process for solving integrals; given a single function to integrate, the typical strategy is to carefully separate this single function into a product of two functions u (x) v (x) such that the residual integral from the integration by parts formula is easier to evaluate than the single function. The following form is useful in illustrating the best strategy to take:

∫ u v d x = u ∫ v d x − ∫ (u ′ ∫ v d x) d x. {\displaystyle \int uv\,dx=u\int v\,dx-\int \left(u'\int v\,dx\right)\,dx.}

On the right-hand side, u is differentiated and v is integrated; consequently it is useful to choose u as a function that simplifies when differentiated, or to choose v as a function that simplifies when integrated. As a simple example, consider:

∫ ln (x) x 2 d x. {\displaystyle \int {\frac {\ln(x)}{x^{2}}}\,dx\,.}

Since the derivative of ln(x) is 1 / x, one makes (ln(x)) part u; since the antiderivative of 1 / x is − 1 / x, one makes 1 / x part v. The formula now yields:

∫ ln (x) x 2 d x = − ln (x) x − ∫ (1 x) (− 1 x) d x. {\displaystyle \int {\frac {\ln(x)}{x^{2}}}\,dx=-{\frac {\ln(x)}{x}}-\int {\biggl (}{\frac {1}{x}}{\biggr)}{\biggl (}-{\frac {1}{x}}{\biggr)}\,dx\,.}

The antiderivative of − 1 / x can be found with the power rule and is 1 / x.

Alternatively, one may choose u and v such that the product u ′ (∫ v dx) simplifies due to cancellation. For example, suppose one wishes to integrate:

∫ sec 2 (x) ⋅ ln (| sin (x) |) d x. {\displaystyle \int \sec ^{2}(x)\cdot \ln {\Big (}{\bigl |}\sin(x){\bigr |}{\Big)}\,dx.}

If we choose u (x) = ln(|sin(x)|) and v (x) = sec x, then u differentiates to 1 tan x {\displaystyle {\frac {1}{\tan x}}} using the chain rule and v integrates to tan x; so the formula gives:

∫ sec 2 (x) ⋅ ln (| sin (x) |) d x = tan (x) ⋅ ln (| sin (x) |) − ∫ tan (x) ⋅ 1 tan (x) d x. {\displaystyle \int \sec ^{2}(x)\cdot \ln {\Big (}{\bigl |}\sin(x){\bigr |}{\Big)}\,dx=\tan(x)\cdot \ln {\Big (}{\bigl |}\sin(x){\bigr |}{\Big)}-\int \tan(x)\cdot {\frac {1}{\tan(x)}}\,dx\.}

The integrand simplifies to 1, so the antiderivative is x. Finding a simplifying combination frequently involves experimentation.

In some applications, it may not be necessary to ensure that the integral produced by integration by parts has a simple form; for example, in numerical analysis, it may suffice that it has small magnitude and so contributes only a small error term. Some other special techniques are demonstrated in the examples below.