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:


C = 0


A – B = 0 A = B


B + A = 1 2A = 1 A = 1/2


A = B = 1/2


Thus I can be expressed as:


I =


I =


Let I1 = and I2 =


I = I1 + I2 ….equation 1


I1 =


Let, u = sin x – cos x du = (cos x + sin x)dx


So, I1 reduces to:


I1 =


I1 = …..equation 2


As, I2 =


I2 = …..equation 3


From equation 1 ,2 and 3 we have:


I =


I =


1