The value of $\int_{0}^{\pi /2} \frac{\sin^{2/3} x}{\sin^{2/3} x + \cos^{2/3} x} dx$ is

If $M=\int_0^{\infty} \frac{\log t}{1+t^3} d t$ and $N=\int_{-\infty}^{\infty} \frac{t e^{2 t}}{1+e^{3 t}} d t$,then

$\int_2^4 \frac{\log x^2}{\log x^2+\log (36-12x+x^2)} dx$ is equal to

For $n \in N$,if $I_n = \int \frac{\sin nx}{\sin x} dx = \frac{2}{n-1} \sin(n-1)x + I_{n-2}$ and $\int_0^\pi \frac{\sin nx}{\sin x} dx = \frac{k\pi}{2}$,then $k =$

The true set of values of $a$ for which the inequality $\int_{a}^{0} (3^{-2x} - 2 \cdot 3^{-x}) \, dx \geq 0$ is true is:

The value of the integral $\int_{-1 / 2}^{1 / 2}\left\{\left(\frac{x+1}{x-1}\right)^{2}+\left(\frac{x-1}{x+1}\right)^{2}-2\right\}^{1 / 2} d x$ is equal to