The value of \int_{0}^{\pi / 2} \frac{d x}{1+\tan ^{3} x} is

NIMCET 2008 Question Paper with solutions

Question : 36 of 120

Marks: +1, -0

The value of

∫↙{0}↖{π / 2} \;{d x}/{1+\tan ^{3} x}

Solution:

I=∫↙{0}↖{π / 2} \;{d x}/{1+\tan ^{3} x}=∫↙{0}↖{π / 2} \;{d x}/{1+\tan ^{3}(\;{π}/{2}-x)}

=∫↙{0}↖{π / 2} \;{d x}/{1+\cot ^{3} x}

⇒ \;\; 2 I=∫↙{0}↖{π \/ 2}(\;{1}/{1+\tan ^{3} x}+\;{1}/{1+\cot ^{3} x}) d x

⇒ \;\; 2 I=∫↙{0}↖{π / 2} d x=\;{π}/{2}

⇒ \;\; I=\;{π}/{4}

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