$\int_0^{\pi /4} \frac{\sec^2 x}{(1 + \tan x)(2 + \tan x)} \,dx = $

$\int_{\frac{1}{25}}^3 \frac{e^{\frac{3}{x}}}{x^2} d x=$

The value of the definite integral,$\int\limits_0^{\sqrt {\ln \left( {\frac{\pi }{2}} \right)} } {\cos \left( {{e^{{x^2}}}} \right)} \cdot 2x {e^{{x^2}}}dx$ is

Let $y=f(x)$ be a thrice differentiable function in $(-5,5)$. Let the tangents to the curve $y=f(x)$ at $(1, f(1))$ and $(3, f(3))$ make angles $\frac{\pi}{6}$ and $\frac{\pi}{4}$,respectively with the positive $x$-axis. If $27 \int_1^3\left(\left(f^{\prime}(t)\right)^2+1\right) f^{\prime \prime}(t) d t=\alpha+\beta \sqrt{3}$,where $\alpha$ and $\beta$ are integers,then the value of $\alpha+\beta$ equals:

Which of the following is/are correct?

$\int_0^3 \frac{dx}{(x+2) \sqrt{x+1}} = $