The value of $k \in N$ for which the integral $I_n = \int_0^1 (1 - x^k)^n dx, n \in N$,satisfies $147 I_{20} = 148 I_{21}$ is :

If $\int \frac{a \cos x+3 \sin x}{5 \cos x+2 \sin x} d x=\frac{26}{29} x-\frac{k}{29} \log |5 \cos x+2 \sin x|+c$,then $|a+k|=$

$\int \left( \frac{x}{x \cos x - \sin x} \right)^2 dx = $

If $\int(\sin x )^{\frac{-11}{2}}(\cos x )^{\frac{-5}{2}} dx = -\frac{p_1}{q_1}(\cot x)^{\frac{9}{2}}-\frac{p_2}{q_2}(\cot x)^{\frac{5}{2}}-\frac{p_3}{q_3}(\cot x)^{\frac{1}{2}}+\frac{p_4}{q_4}(\cot x)^{\frac{-3}{2}}+C,$ where $p_i$ and $q_i$ are positive integers with $\operatorname{gcd}(p_i, q_i)=1$ for $i =1,2,3,4$ and $C$ is the constant of integration,then $\frac{15 p_1 p_2 p_3 p_4}{q_1 q_2 q_3 q_4}$ is equal to . . . . . . .

$\int \frac{\sin x+8 \cos x}{4 \sin x+6 \cos x} d x$ is equal to

If $f(x) = \int \operatorname{cosec}^5 x \, dx$,then $f\left(\frac{\pi}{4}\right) = $