$\int_{-\pi}^{\pi} (1-x^2) \sin x \cdot \cos^2 x \, dx$ is equal to:

If $M = \int_{0}^{\pi / 2} \frac{\cos x}{x+2} dx$ and $N = \int_{0}^{\frac{\pi}{4}} \frac{\sin x \cos x}{(x+1)^{2}} dx$, then the value of $M-N$ is

Let $a_n = \int_{-\pi}^{\pi} |x-1| \cos(nx) \, dx$ for all natural numbers $n$. Then,the sequence $(a_n)_{n \geq 1}$ satisfies:

The value of the integral $\int_{-\pi/4}^{\pi/4} \left( \frac{32 \cos^4 x}{1 + e^{\sin x}} \right) dx$ is:

The value of $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{x^2 \cos x}{1+e^x} d x$ is equal to

Assertion $(A): \int_{-a}^a f(x) dx = \int_0^a (f(x) + f(-x)) dx$
Reason $(R): \int_a^b f(x) dx = \int_{g(a)}^{g(b)} f(g(u)) g'(u) du$
The correct option among the following is: