$\int \frac{dx}{(e^x - 1)} =$

$\int \frac{1}{x(x^5-1)} dx$ is equal to

The value of the definite integral $I = \int_{-1}^{1} \frac{1}{1 + \sqrt{e^x}} dx$ is:

Let $f(x)=\int \frac{3x-2}{(x+1)^2(x+3)}dx = \alpha\log\left|\frac{x+1}{x+3}\right|+\beta\left(\frac{1}{x+1}\right)+\gamma$, where $\gamma$ is the constant of integration. If $f(0)=\frac{11}{4}\log\left(\frac{1}{3}\right)+4$, then which of the following are correct?

(A) $4\alpha+2\beta+6\gamma=22$

(B) $f(1)=\alpha[1-\log 2]$

(C) $\alpha+\beta+\gamma=\frac{27}{4}$

(D) $\gamma=-\frac{5}{2}$

Choose the correct answer from the options given below:

$\int \frac{dx}{x(x^{10}+1)}$ is equal to

The value of $\int_{0}^{\pi/2} \frac{\tan^7 x}{\cot^7 x + \tan^7 x} dx$ is

If $\dfrac{d}{dx}[f(x)] = 5x^4 - \dfrac{2}{x^3}$ such that $f\left(-\dfrac{1}{2}\right) = 0$, then $f(x)$ is equal to