Lesson Integration by parts (Calculus)

In integral calculus, integration by parts is a technique used to evaluate the integral of a complex expression, typically a product of two functions that do not work well together (e.g. a polynomial function with a trigonometric function). Here, we cover the basics of integration by parts, with an example.

Recall that the product rule says that, for two functions u and v (both in terms of x),



We can integrate both sides to obtain

, which implies that

. This is the integration by parts formula. Note that u, v, du, and dv are in terms of x and dx.

Example
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Integrate:
.

We want to use a series of substitutions so that one "piece" of the integral represents a function u, and the other "piece" represents the derivative of another function, hence dv. A general rule of thumb is to let u be an expression that is easy to differentiate, and results in a simpler expression. Also, you want v to be an expression that is easy to integrate.

Suppose we let . Then, we can let . From this, we can find du (the derivative of u), as well as v (by integrating dv).




Applying the integration by parts formula and substituting,





Here, we get the integral of -cos(x) dx, which is basic:

, where C is an arbitrary constant.