substitution rule: evaluate the indefinite integral

Question

a) integral sinx/(1+cos^2 x) dx

b) integral (1-4x)/(sqrt(1-4x^2)) dx

c) integral (lnx+4)/(x(lnx)^3) dx

d) integral x^2(sqrt(2+x)) dx

e) integral 1 to 4   (e^(sqrt x)+1)/(sqrt x) dx

f) integral 0 to 15   xsqrt(36+3x)  dx

Answer 1

(a)

Step 1 :

.

Consider .

Differentiate with respect to .

Step 2 :

Substiute and in .

Use integration Formula : +c.

+c

Substitute in the above expression.

+c.

+c.

Solution :

+c.

 

Answer 2

(e)

Step 1:

The integral is .

Consider .

Substitute and in the integral.

Repalce in the above expression.

                       

.

Solution :

.

Answer 3

(f)

Step 1:

The integral is .

Consider .

Substitute and in the integral.

Repalce in the above expression.

                          

.

Solution :

.

Answer 4

(b)

Step 1 :

.

Use sum rule of integration : .

.

Step 2 :

Consider .

Differentiate with respect to .

.

Use power rule of derivative : .

Substitute and   in .

Use integration Formula : .

Substitute .

.

Comments

Step 3 :

.

Consider .

Differentiate  .

Use power rule of derivative : .

Substitute and .

Substitute .

.

 .

Step 4 :

Substitute and in

.

 

.

Solution :

.

Answer 5

(c)

Step 1:

The integral is .

Sum rule of integration: .

Consider .

Assume .

Differentiate with respect to .

Substitute and in the integral.

Power rule of integration: .

Replace in above expression.

.

Comments

Continued....

Step 2:

Consider .

Assume

Differentiate with respect to .

Substitute and in the integral.

Replace in above expression.

.

 

Substitute the results of  and in .

.

Solution:

.

Answer 6

(d)

Step 1: 

The integral is .

Assume .

Differentiate with respect to .

Substitute and in the integral.

Power rule of integration: .

.

Replace in above expression.

.

Solution:

.