Integration By Parts Formula - Practice Questions & MCQ

To evaluate integration of a function which is a product of two functions, we use the method of integration by parts.

If two functions of x, u and v are given, then

i.e.

The integral of product of two functions =

( first function) X ( integral of second function ) - integral of (differential of first function X integral of second function).

Proof:

Let, h(x) = f(x) . g(x), then by using the product rule, we obtain, h′(x) = f ′(x) g(x) + g′(x) f(x). Let’s now integrate both sides of this equation:

Important point for selecting first function (u) and second function (v)

Usually we use the following preference order for selecting the first function. (Inverse, Logarithmic, Algebraic, Trigonometric, Exponential). In the above stated order, the function on the left is always chosen as the first function. This rule is called as ILATE.

For example, if an integral contains a logarithmic function and an algebraic function, we should choose logarithmic function as the first function (u). 

One of the interesting application of integration by part is:

Proof:

General Formula

In the same way,