help? partial fractions-integrals

Question

Answer 1

(1)

Step 1:

The integral is .

Use partial fraction method :

Substitute t = 12.

Substitute t = 0.

So .

Step 2:

Sum property of Integral :.

Integral of logarithm function is .

.

Solution :

.

Answer 2

(2)

Step 1:

The integral is .

Use partial fraction method :

Consider denominator

Substitute .

Substitute .

So .

Step 2:

Sum property of Integral :.

Integral of logarithm function is .

.

Solution :

.

Answer 3

(3)

Step 1:

The integral is .

Use partial fraction method :

Substitute .

Substitute .

So .

Step 2:

Sum property of Integral :.

Integral of logarithm function is .

.

Solution :

.

Answer 4

(4)

Step 1:

The integral is .

Use partial fraction method :

Substitute .

Compare the coefficient of .

Compare the coefficient of .

So .

Step 2:

Sum property of Integral :.

Integral of logarithm function is .

Use the integral of inverse tangent function : .

.

 

Solution :

.

Answer 5

(5)

Step 1:

The integral is .

Consider .

Rewrite the fraction as :

.

Long division method :

Step 2:

Consider .

Use partial fraction method :

Substitute in above equation (1).

Substitute in above equation.

Compare the constant terms.

Step 3: 

So .

Apply sum property of integral :.

Apply constant rule of integration : .

Apply integral of logarithm function : .

Solution:

.