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

$\displaystyle\int e^{x\log_e 7} e^x\, dx$ is equal to :

[where C is an arbitrary constant]

The value of $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} (\sin|x| + \cos|x|)dx$, is equal to:

The value of the integral $\int_{-6}^{0} |x+3|\,dx$ is

If $\int (1 + e^{-x} + e^{-2x} + ...)dx = \log\phi(x) + C$, then $\phi(x)$ is equal to :

If the integral $I = \int \frac{x^2}{\sqrt{1+x}} dx = \frac{1}{\alpha} (1+x)^\alpha - \frac{8\alpha}{15} (1+x)^{\alpha-1} + 2(1+x)^{\alpha-2} + C$, $C$ is constant of integration, then the value of $\alpha$ is:

The value of the definite integral $I = \int_{1}^{2} \frac{1}{x(1 + x^2)}dx$ is: