If the integral $\int \frac{\cos 8x + 1}{\cot 2x - \tan 2x} dx = A \cos 8x + k,$ where $k$ is an arbitrary constant,then $A$ is equal to

If $\int(x^6+x^4+x^2) \sqrt{2x^4+3x^2+6} dx = f(x) + c$,then $f(3) =$

If $\int e^x \left(f(x) - f^{\prime}(x)\right) dx = g(x) + C$,then $\int e^x f^{\prime}(x) dx =$

The value of $\frac{\int_0^{\pi / 2}(\sin x)^{\sqrt{2}+1} d x}{\int_0^{\pi / 2}(\sin x)^{\sqrt{2}-1} d x}$ is $........$

If $\int \frac{(\sqrt{1+x^2}+x)^{10}}{(\sqrt{1+x^2}-x)^9} dx = \frac{1}{m}((\sqrt{1+x^2}+x)^n (n\sqrt{1+x^2}-x)) + C$,where $C$ is the constant of integration and $m, n \in N$,then $m+n$ is equal to

If $\int \frac{dx}{x^{2022}(1+x^{2022})^{1/2022}} = \frac{-(1+x^m)^{n/m}}{nx^n} + C$,then $m-n=$