Definite Integration Question 479
Question: The value of $ \int_0^{\pi /2}{\frac{dx}{1+{{\tan }^{3}}x}} $ is
[IIT 1993; DCE 2000, 01]
Options:
A) 0
B) 1
C) $ \frac{\pi }{2} $
D) $ \frac{\pi }{4} $
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Answer:
Correct Answer: D
Solution:
$ I=\int_0^{\pi /2}{\frac{dx}{1+{{\tan }^{3}}x}=\int_0^{\pi /2}{\frac{{{\cos }^{3}}x}{{{\sin }^{3}}x+\cos x^{3}}}}dx $ …..(i) $ =\int_0^{\pi /2}{\frac{{{\sin }^{3}}x}{{{\cos }^{3}}x+{{\sin }^{3}}x}dx} $ ……(ii)
Adding (i) and (ii), we get $ 2I=\int_0^{\pi /2}{dx\Rightarrow I=\frac{\pi }{4}.} $
BETA
