Q. 105.0( 1 Vote )

Evaluate the integral


Answer :

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


2B - 3A = 1


3B + 2A = 8


On solving for A ,B and C we have:


A = 1 , B = 2 and C = 0


Thus I can be expressed as:


I =


I =


Let I1 = and I2 =


I = I1 + I2 ….equation 1


I1 =


Let, 3 cos x + 2 sin x = u


(-3sin x + 2cos x)dx = du


So, I1 reduces to:


I1 =


I1 = …..equation 2


As, I2 =


I2 = …..equation 3


From equation 1, 2 and 3 we have:


I =


I =


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