Evaluate the integral
Ideas required to solve the problems:
* Integration by substitution: A change in the variable of integration often reduces an integral to one of the fundamental integration. If derivative of a function is present in an integration or if chances of its presence after few modification is possible then we apply integration by substitution method.
* Knowledge of integration of fundamental functions like sin, cos ,polynomial, log etc and formula for some special functions.
Let, I = 
To solve such integrals involving trigonometric terms in numerator and denominators. We use the basic substitution method and to apply this simply we follow the undermentioned procedure-
If I has the form 
Then substitute numerator as -
Where A, B and C are constants
We have, I = 
As I matches with the form described above, So we will take the steps as described.
∴
⇒
{
⇒ 
Comparing both sides we have:
3B+ C = 3
B + 2A = 2
2B - A = 4
On solving for A ,B and C we have:
A = 0, B = 2 and C = -3
Thus I can be expressed as:
I = 
I = 
∴ Let I1 = 
and I2 = 
⇒ I = I1 + I2 ….equation 1
I1 = 
So, I1 reduces to:
I1 = 
…..equation 2
As, I2 = 
To solve the integrals of the form 
To apply substitution method we take following procedure.
We substitute:
∴ I2 = 
⇒ I2 = 
⇒ I2 = 
⇒ I2 = 
Let, t = 
⇒ 
∴ I2 = 
As, the denominator is polynomial without any square root term. So one of the special integral will be used to solve I2.
I2 = 
⇒ I2 = 
∴ I2 = 
{∵ a2 + 2ab + b2 = (a+b)2}
As, I2 matches with the special integral form
I2 = 
Putting value of t we have:
∴ I2 = 
+ C2 ……equation 3
From equation 1,2 and 3:
I = 
+ C2
∴ I = 
+ C ….ans