If $\int {\frac{1}{{x + {x^5}}}dx = f(x) + c} $,then the value of $\int {\frac{{{x^4}}}{{x + {x^5}}}dx} $ is

If $\int \operatorname{cosec}^5 x \, dx = \alpha \cot x \operatorname{cosec} x \left(\operatorname{cosec}^2 x + \frac{3}{2}\right) + \beta \log_e \left|\tan \frac{x}{2}\right| + C$,where $\alpha, \beta \in R$ and $C$ is the constant of integration,then the value of $8(\alpha + \beta)$ is equal to:

If $\int \frac{\sin x}{\sin (x-\alpha)} dx = px - q \log |\sin (x-\alpha)| + c$,then $pq =$ . . . . . . .

If $\int \frac{d \theta}{\cos ^2 \theta(\tan 2 \theta+\sec 2 \theta)}=\lambda \tan \theta+2 \log _{e}|f(\theta)|+c$ (where $c$ is a constant of integration),then the ordered pair $(\lambda, |f(\theta)|)$ is equal to

$\int \frac{6x+5}{\sqrt{6+x-2x^2}} dx =$

If $\int \frac{2 x^{12}+5 x^9}{\left(1+x^3+x^5\right)^3} d x=\frac{x^m}{l\left(1+x^3+x^5\right)^r}+C$ then $\frac{m-l}{r}=$