Integration by Parts

Let u and v be two differentiable function of a single independent variable x.

Integrating w.r.t x on both sides

Let u = f(x),

Then

\ (1) can be written as

Note:

The above can be remembered as

(First function) (Integral of Second function)

Note:

(1) While integration by parts, the proper choice of first function and second function is significant

Try integrating

Taking x as the first function and e^x as the second function is simpler than taking x as the second function.

(2) Integrating by parts may not be applicable to product of functions in all cases. In some cases the product of two functions may not be integrable.

(3) While finding the integral of the second function we do not add constant of integration.

We need not add a constant of integration to the second function as it gets cancelled in the final results.

(4) Some times even if the integral is not a product of two functions, the method of integration by parts can be used.

Example: Let us integrate sin^-1x

where

Example:

Integrate the following function (x² + 1) logx

Suggested answer:

Integrating by parts

Integral of the form

Method:

Put e^x f(x) = t

Differentiating w.r.t. x,

{e^xf(x) +e^xf^'(x)} dx = dt

e^x{f(x) + f^'(x)} dx = dt

= t + c

= e^xf(x)+c

Evaluate the integral .

Suggested answer:

Differentiating,