If $I_n = \int \frac{\sin nx}{\cos x} dx$,then $I_n =$

$\int \frac{1}{x^5 \sqrt[5]{x^5+1}} d x=$

If $\int \frac{d x}{\cot ^2 x-1}=\frac{1}{A} \log |\sec 2 x+\tan 2 x|-\frac{x}{B}+c$,(where $c$ is the constant of integration),then $A+B=$

If $\int \frac{7 x^8+8 x^7}{\left(1+x+x^8\right)^2} d x=f(x)+c$, then $f(x)$ is equal to

Let $I(x)=\int\frac{3dx}{(4x+6)(\sqrt{4x^{2}+8x+3})}$ and $I(0)=\frac{\sqrt{3}}{4}+20$. If $I(\frac{1}{2})=\frac{a\sqrt{2}}{b}+c$, where $a, b, c \in N$ and $gcd(a,b)=1$, then $a+b+c$ is equal to:

Let $I(x) = \int \frac{dx}{(x-11)^{\frac{11}{13}}(x+15)^{\frac{15}{13}}}$. If $I(37) - I(24) = \frac{1}{4} \left( \frac{1}{b^{\frac{1}{13}}} - \frac{1}{c^{\frac{1}{13}}} \right)$,where $b, c \in \mathbb{N}$,then $3(b+c)$ is equal to