Value of $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \log(\tan x)dx$ is

If $\int_0^{\pi/2} \sqrt{\tan x} \, dx = \frac{\lambda}{\sqrt{2}}$, then the value of $\lambda$ is

Value of $\int_{1}^{4} \frac{\log_e x}{x^2}\, dx$ is

If $f(x) = \begin{cases} 1 - 2x, & x \leq 0 \\ 1 + 2x, & x > 0 \end{cases}$, then $\int_{-1}^{1} f(x) \, dx =$

The value of $\int_{0}^{1} \frac{a-b x^{2}}{\left(a+b x^{2}\right)^{2}} d x$ is :

For $x \in \left(0, \frac{\pi}{2}\right)$, $\int \frac{1}{\sin^2 x + \sin 2x} dx$ is equal to

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

[where C is an arbitrary constant]