JEE PYQ: Indefinite Integration Question 15
Question 15 - 2020 (08 Jan Shift 1)
If $\int \frac{\cos x,dx}{\sin^3 x(1+\sin^6 x)^{2/3}} = f(x)(1+\sin^6 x)^{1/\lambda} + c$ where $c$ is a constant of integration, then $\lambda f\left(\frac{\pi}{3}\right)$ is equal to:
(1) $-\frac{9}{8}$
(2) 2
(3) $\frac{9}{8}$
(4) $-2$
Show Answer
Answer: (4)
Solution
Let $\sin x = t$, $\cos x,dx = dt$. $I = \int \frac{dt}{t^3(1+t^6)^{2/3}}$. Put $1 + t^{-6} = r^3$, then $\frac{dt}{t^7} = -\frac{1}{2}r^2,dr$. After substitution: $f(x) = -\frac{1}{2\sin^2 x}$, $\lambda = 3$. So $\lambda f\left(\frac{\pi}{3}\right) = 3 \cdot \left(-\frac{1}{2}\right)\operatorname{cosec}^2\frac{\pi}{3} = 3 \cdot \left(-\frac{1}{2}\right)\cdot\frac{4}{3} = -2$.
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