Indefinite Integration 1 Question 36
37. The value of $\int _0^{\pi / 2} \frac{d x}{1+\tan ^{3} x}$ is
(1993, 1M)
(a) 0
(b) 1
(c) $\pi / 2$
(d) $\pi / 4$
Show Answer
Solution:
- Let $I=\int _0^{\pi / 2} \frac{1}{1+\tan ^{3} x} d x=\int _0^{\pi / 2} \frac{1}{1+\frac{\sin ^{3} x}{\cos ^{3} x}} d x$
$$ \Rightarrow \quad I=\int _0^{\pi / 2} \frac{\cos ^{3} x}{\cos ^{3} x+\sin ^{3} x} d x $$
$\Rightarrow \quad I=\int _0^{\pi / 2} \frac{\cos ^{3} \frac{\pi}{2}-x}{\cos ^{3} \frac{\pi}{2}-x+\sin ^{3} \frac{\pi}{2}-x} d x$
$\Rightarrow \quad I=\int _0^{\pi / 2} \frac{\sin ^{3} x}{\sin ^{3} x+\cos ^{3} x} d x$
On adding Eqs. (i) and (ii), we get
$$ 2 I=\int _0^{\pi / 2} 1 d x \Rightarrow 2 I=[x] _0^{\pi / 2}=\pi / 2 \Rightarrow I=\pi / 4 $$
