Q. 4

Evaluate the following definite Integrals:  [CBSE 2014]

Answer :

We are asked to calculate


For this we have to apply integration by parts


Let u and v be two functions then



To choose the first function u we use “ILATE” rule


That is


I=inverse trigonometric function


L=logarithmic function


A=algebraic function


T=trigonometric functions


E=exponential function


So in this preference, the first function is choosen to make the integration simpler.


Now, In the given question x is an algebraic function and it is chosen as u (A comes first in “ILATE” rule)


So first let us integrate the equation and then let us substitute the limits in it



Therefore, now substitute the limits given:


Note that and




First we have to substitute the upper limit and then subtract the second limit value from it
)


Note that sin0= 0 and cos0=1


=0+1+0–0


=1

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