If ∫ cos x d x sin 3 x 1 + sin 6 x 2 3 = f x 1 + sin 6 x 1 λ + c , where c is a constant of…

Let, $\mathrm{s}\mathrm{i}\mathrm{n}x=t$ 

$\mathrm{c}\mathrm{o}\mathrm{s}xdx=dt$
$I=\int \frac{dt}{t^3{\left(1+t^6\right)}^{\frac{2}{3}}}=\int \frac{dt}{t^7{\left(1+\frac{1}{t^6}\right)}^{\frac{2}{3}}}$

Put $1+\frac{1}{t^6}=r^3$ 

$\Rightarrow \frac{dt}{t^7}=\frac{-1}{2}{r}^{2}dr$

$I=-\frac{1}{2}\int \frac{r^{2}dr}{r^2}=-\frac{1}{2}r+c$ 

$\Rightarrow I=-\frac{1}{2}{\left(\frac{{\sin }^{6}x+1}{{\sin }^{6}x}\right)}^{\frac{1}{3}}+c$

$\Rightarrow I=-\frac{1}{2{\sin }^{2}x}{\left(1+{\sin }^{6}x\right)}^{\frac{1}{3}}+c$

So, $f\left(x\right)=-\frac{1}{2}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{e}{\mathrm{c}}^{2}x$ and $\lambda =3$

$\lambda f\left(\frac{\pi }{3}\right)=-2$