$\int \sqrt{1 + x^2} \, dx = $

If $\int \frac{5 \tan (x)}{\tan (x)-2} d x = x + a \log |\sin (x) - 2 \cos (x)| + k$,then $a$ is equal to

If $\int e^{x+\tan^{-1} x} \left(\frac{x^2+2}{1+x^2}\right) dx = e^{f(x)} + c$, then which of the following is true?

If $\int \frac{(x-1) dx}{(x+1) \sqrt{x^3+x^2+x}} = A \cdot \tan^{-1} \sqrt{f(x)} + \text{constant}$, then the ordered pair $(A, f(-1)) =$

Evaluate the integral: $\int \sec^5 x \, dx$

$\text{If } \int \frac{1}{\operatorname{cosec} x+\cos x} d x = \frac{1}{2 \sqrt{3}} \log |f(x)| - \int \frac{\cos x-\sin x}{2+\sin 2 x} d x + c, \text{ then at } x = \frac{\pi}{3}, |f(x)| = $