Question

Evaluate the integral.

Answer 1

Step 1:

The integral is .

Find the integration using the integration by parts.

Formula for integration by parts:

Consider and .

Power rule of integration:

Power rule of derivatives: and .

.

Step 2:

Consider

Evaluate the result again by using integration by parts.

Assume and .

Find by integrate .

Above expression is of the form , since derivative of  is .

Substitute and in integration by parts.

Answer 2

Continued:

Step 3:

Consider .

Again find the result by considering integration by parts.

Assume and .

                           (Since )

                           (Since )

.

Step 4:

Substitute value in .

Substitute in .

.

Solution:

.