help? [integrals]?

Question

Answer 1

(5)

Step 1:

The integral is .

Sum and difference rule of integral : .

Integral formula of sine function : .

Integral formula of exponential function : .

                            

.

Solution :

.

Answer 2

(6)

Step 1:

The integral is .

Rewrite the integral.

Consider .

Substitute and in the above integral.

                        

Power rule of integral formula : .

                       

Replace in the above expression.

                       

.

Solution :

.

Answer 3

(7)

Step 1:

The integral is .

Consider .

Substitute and  in the above integral.

Power rule of integral formula : .

                      

Replace in the above expression.

                      

.

Solution :

.

Answer 4

(8)

Step 1:

The integral is .

Rewrite the integral.

Consider the function .

Now use the partial fraction method :

Compare the coefficients of x terms.

                    (1)

Compare the constant terms.

            (2)

Multiple equation (1) with 3.

             (3)

Slove equation (2) and (3).

.

Substitute in equation (1).

.

Step 2:

Integral of logarithm function : .

                             

.

Solution :

.