$\int 2x \cos^3(x^2) \sin(x^2) \, dx = $

If $n$ is a natural number, then $\int \frac{\sin^n x}{\cos^{n+2} x} dx =$

Evaluate the integral $\int \frac{\sqrt{\cot x}}{\sin x \cos x} d x = -f(x) + c$. Find $f(x)$.

If $x \in \left( \frac{\pi}{4}, \frac{3\pi}{4} \right)$,then $\int \frac{\sin x - \cos x}{\sqrt{1 - \sin 2x}} e^{\sin x} \cos x \, dx = $

Consider the following statements:
Assertion $(A):$ $\frac{1}{x^2 + a^2}$ can be integrated by a substitution $x = a \tan \theta$.
Reason $(R):$ Because all integrands are integrated by the method of substitution only.
Which of the following is correct?

$\int \frac{\sin(2x)}{\sin^2(x) + 2\cos^2(x)} dx = $