Question

If $\int \frac{\cos x}{{\sin }^{3}x(1+{\sin }^{6}x{)}^{2/3}}dx=f\left(x\right)(1+{\sin }^{6}x{)}^{1/\lambda }+c$, where $c$ is a constant of integration, then $\lambda f\left(\frac{\pi }{3}\right)$ is equal to :

• A. $-\frac{9}{8}$
• B. $\frac{9}{8}$
• C. $2$
• D. $-2$

Answer 1

The correct option is D $-2$

Let $\sin x=t\Rightarrow \cos xdx=dt$
$\therefore \int \frac{dt}{t^3(1+t^6{)}^{2/3}}=\int \frac{dt}{t^7{(1+\frac{1}{t^6})}^{2/3}}$

Let $1+\frac{1}{t^6}=u\Rightarrow -6t^{-7}dt=du$
$\therefore \int \frac{dt}{t^7{(1+\frac{1}{t^6})}^{2/3}}$
$=-\frac{1}{6}\int \frac{du}{u^{2/3}}$
$=-\frac{3}{6}{u}^{1/3}+c=-\frac{1}{2}{(1+\frac{1}{t^6})}^{1/3}+c$
$=-\frac{(1+{\sin }^{6}x{)}^{1/3}}{2{\sin }^{2}x}+c=f\left(x\right)(1+{\sin }^{6}x{)}^{1/\lambda }+c$
$\therefore \lambda =3$ and $f\left(x\right)=-\frac{1}{2{\sin }^{2}x}$
$\Rightarrow \lambda f\left(\frac{\pi }{3}\right)=-2$