help---[integration by parts]???

Question

Answer 1

(1)

Step 1 :

The integration function is.

Let .

Differentiate with respect to .

.

Substitute in equation (1).

Step 2 :

Integration by parts: .

 Let and .

Differentiate with respect .

Substite and in above formula.

Integration of exponential function :.

Substitute in equation (2).

.

Solution :

.

Answer 2

(3)

Step 1:

The integral is.

Determine the integral by using integration by parts.

Integration by parts: .

Let and .

Find by integrating .

.

.

Differentiate on each side.

.

Substitute corresponding values in integration by parts.

Step 2:

Consider .

Integration by parts: .

Let and .

Find by integrating .

.

Differentiate on each side.

.

Substitute corresponding values in integration by parts.

Comments

Continued.....

Substitute the result of in .

 

.

Solution:

.

Answer 3

2)

Step 1: 

The integral is.

Determine the integral by using integration by parts.

Integration by parts: .

Let and .

Find by integrating .

.

.

Differentiate on each side.

.

Step 2:

Substitute corresponding values in integration by parts.

.

Solution:

.

Answer 4

(4)

Step 1 :

The integration function is .

Determine the integral by using integration by parts.

Integration by parts: .

Let  and .

Differentiate with respect to .

Integrate on both sides.

Substitute , ,  and .

Step 2 :

Consider.

Let  and .

Differentiate with respect to .

.

Comments

Continued....

Substitute , ,  and .

Substitute equation (2) in equation (1).

Solution :

.