MHT CET · Maths · Definite Integration
\(\frac{\pi}{2}\)
The value of \(\int_0 \frac{\mathrm{d} x}{1+\tan ^3 x}\)
- A 0
- B \(\frac{\pi}{4}\)
- C \(\frac{\pi}{2}\)
- D 1
Answer & Solution
Correct Answer
(B) \(\frac{\pi}{4}\)
Step-by-step Solution
Detailed explanation
\(\int_0^{\frac{\pi}{2}} \frac{\mathrm{d} x}{1+\tan ^3 x}=\int_0^{\frac{\pi}{2}} \frac{\cos ^3 x}{\cos ^3 x+\sin ^3 x} \mathrm{~d} x=\) \(\frac{\frac{\pi}{2}-0}{2}=\frac{\pi}{4}\)
\({\left[\because \int_a^b \frac{f(x) \mathrm{d} x}{f(x)+f(a+b-x)}=\frac{b-a}{2}\right]}\)
\({\left[\because \int_a^b \frac{f(x) \mathrm{d} x}{f(x)+f(a+b-x)}=\frac{b-a}{2}\right]}\)
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