The value of the integral $I=\displaystyle\int\dfrac{e^x(x-2)}{(x+1)^4}dx$ is, [$C$ is an arbitrary constant]

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If $I = \int \frac{x^4 + x^2 + 1}{x^2 - x + 1} dx = \alpha x + \beta x^2 + \gamma x^3 + \delta$, $\delta$ is constant of integration, then $(\alpha + 2\beta + 3\gamma)$ equals

$\displaystyle\int_0^2 x\sqrt{2-x}\,dx$ is equal to :

$\int e^{x\log_e a}.e^x dx$ is:

[where C is an arbitrary constant]

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

$\displaystyle\int \dfrac{x}{x-\sqrt{x^2-1}}\,dx$ is equal to :