If $f(x)$ and $g(x)$ are continuous functions in [0, a] such that $f(x) = f(a - x)$ and $g(x) + g(a - x) = a$ then $\int_{0}^{a} f(x)g(x)dx =$

Let $[t]$ denotes the greatest integer $\le t$, then the value of $\left[\displaystyle\int_{\pi}^{3\pi/2} x|\sin x|\,dx\right]$ is:

The integral $\int \frac{dx}{x^2(x^4+1)^{\frac{3}{4}}}$ equals __________.

The value of $\int_{0}^{3} \frac{\sqrt{x}}{\sqrt{3-x}+\sqrt{x}}\,dx$ is

$\int \sin x \sin 2x \sin 3x dx$ is equal to

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

(where $c$ is an arbitrary constant)

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