Q17 of 43 Page 1

Evaluate: .

OR


Evaluate:

dx


∵ - x = x( - 1) = x (x – 1) (x + 1)


- x > 0 for -1 < x < 0


- x < 0 for 0 < x < 1


- x > 0 for 1 < x < 2


NOTE: Sign is only being considered for the range -1≤ x ≤ 2


dx + dx + dx


dx – dx + dx


I = - +


I =


I =


The required value is .


OR


Let, I = …(1)


Using the property of definite Integral: =


I =


I = …(2)


Adding equation 1 and 2 we get –


2I =


2I =


Putting Cos x = t


-Sin x dx = dt


When x = 0 ⇒ t = 1


x = ⇒ t = -1


⇒ 2I = {∵ }


⇒ 2I = π


∴ 2I = π{tan-1(1) – tan-1(-1)} {∵ tan-1(1) = π/4}


⇒ 2I = π(π/4 – (-π/4))


⇒ I =


Hence,


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