Evaluate $\int_{\sin \theta}^{\cos \theta} f(x \tan \theta) \, dx$,where $\theta \neq \frac{n \pi}{2}, n \in I$.

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:

$\int_{-3}^0 x \sqrt{x+4} \, dx =$

Let $\frac{d}{dx}F(x) = \frac{e^{\sin x}}{x}$ for $x > 0$. If $\int_{1}^{4} \frac{3}{x} e^{\sin(x^3)} dx = F(k) - F(1)$,then one of the possible values of $k$ is:

$\int_{\pi /4}^{\pi /2} \cos \theta \csc^2 \theta \, d\theta = $

$\int_0^{\pi / 4} \frac{\cos ^2 x \sin ^2 x}{\left(\cos ^3 x+\sin ^3 x\right)^2} d x$ is equal to