The value of $\int_0^{\pi/2} \frac{\cos x}{\cos\frac{x}{2}+\sin\frac{x}{2}}\,dx$ is

If $\displaystyle\int\dfrac{x^3\,dx}{\sqrt{1+x^2}} = a(1+x^2)^{3/2}+b\sqrt{1+x^2}+c$, where 'C' is an arbitrary constant, then the value of $a$ and $b$ are

$\int_{0}^{\pi} \frac{e^{\cos x}}{e^{\cos x} + e^{-\cos x}} dx =$

$\int_0^{\pi/2} \sqrt{1 - \sin 2x} \, dx$ is equal to :

The integral $\int e^x\left(\frac{x-1}{2x^2}\right)dx$ is equal to

$\int e^{-x}(\cot x + \cosec^2 x)dx =$

If $\int (x + \sqrt{x^2 - 1})^2 \, dx = \alpha \cdot x + \beta x^3 + \gamma (x^2 - 1)^{\frac{3}{2}} + C$, where $C$ is arbitrary constant, then the value of $3(\alpha + \beta + \gamma)$ is