How to evaluate the following items in mathematical methods in physics? [closed]

HINTS:

For $1)$, write

$$\sum_{n=0}^\infty p^n\sin(2nx)=\text{Im}\left(\sum_{n=0}^\infty (p\,e^{i2x})^n\right)\tag1$$

Then sum the geometric series on the right-hand side of $(1)$ and take the imaginary part of that result.

For $3)$, note that

$$\int_0^\pi \frac{1}{(a+\cos(\theta))^2}\,d\theta=-\frac{d}{da}\int_0^\pi \frac{1}{a+\cos(\theta)}\,d\theta\tag 2$$

The integral on the right-hand side of $(2)$ can be evaluated using the classical tangent half-angle substitution or contour integration. Finish by taking the derivative with respect to $a$ and multiplying by $-1$.

For $4)$ note that

$$\int_0^\infty x^6e^{-x^2}\,dx=-\left.\left(\frac{d^3}{da^3}\int_0^\infty e^{-ax^2}\,dx\right)\right|_{a=1} \tag 3$$

The integral on the right-hand side of $(3)$ is the form of the classical Gaussian integral. Take its third derivative, evaluate at $a=1$, and multiply by $-1$.