Let $2f(x) + f(-x) = \frac{1}{x} \sin \left( x - \frac{1}{x} \right)$. Then the value of $\int_{1/e}^{e} f(x) dx$ is -

The value of $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{1}{[x]+4} dx$, where $[\bullet]$ denotes the greatest integer function, is

If $(a, b)$ is the orthocentre of the triangle whose vertices are $(1, 2), (2, 3)$ and $(3, 1)$,and $I_1 = \int_{a}^{b} x \sin(4x - x^2) dx$,$I_2 = \int_{a}^{b} \sin(4x - x^2) dx$,then $36 \frac{I_1}{I_2}$ is equal to:

If $\int_{0}^{2}(\sqrt{2x}-\sqrt{2x-x^{2}}) dx = \int_{0}^{1}(1-\sqrt{1-y^{2}}-\frac{y^{2}}{2}) dy + \int_{1}^{2}(2-\frac{y^{2}}{2}) dy + I$,then $I = \dots$

If the value of the integral $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\frac{x^2 \cos x}{1+\pi^x}+\frac{1+\sin ^2 x}{1+e^{\sin x^{323}}}\right) d x=\frac{\pi}{4}(\pi+a)-2$,then the value of $a$ is

Let $(a, b)$ be the point of intersection of the curve $x^2=2y$ and the straight line $y-2x-6=0$ in the second quadrant. Then the integral $I=\int_a^b \frac{9x^2}{1+5^x} dx$ is equal to: