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The value of int(0)^(pi//2)(dx)/(1+tan^(...

The value of `int_(0)^(pi//2)(dx)/(1+tan^(3)x)` is

A

0

B

1

C

`pi//2`

D

`pi//4`

Text Solution

Verified by Experts

The correct Answer is:
D
Let `I=int_(0)^(pi//2)(1)/(1+tan^(3)x)dx=int_(0)^(pi//2)(1)/(1+(sin^(3)x)/(cos^(3)x))dx`
`rArrI=int_(0)^(pi//2)(cos^(3))/(cos^(3)x+sin^(3)x)dx` . . . (i)
`rArrI=int_(0)^(pi//2)cos^(3)(pi/(2)-x)/(cos^(3)((pi)/(2)-x)+sin^(3)((pi)/(2)-x))dx`
`rArr I=int_(0)^(pi//2)(sin^(3))/(sin^(3)x+cos^(3)x)dx` . . . (ii) On adding Eqs . (i) and (ii) , we get
`2I=int_(0)^(pi//2)1dxrArr2I=[x]_(0)^(pi//2)=pi//2rArrI=pi//4`
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