The value of $\int_{0}^{1} 9x^8 dx + \int_{0}^{\pi/2} \cos x dx$ is

If $\int_0^{2 \pi}\left(\sin ^4 x+\cos ^4 x\right) d x=K \int_0^\pi \sin ^2 x d x+L \int_0^{\frac{\pi}{2}} \cos ^2 x d x$ and $K, L \in N$,then the number of possible ordered pairs $(K, L)$ is

Let $a, b, c$ be non-zero real numbers such that $\int_0^1 {(1 + \cos^8 x)(ax^2 + bx + c) \, dx} = \int_0^2 {(1 + \cos^8 x)(ax^2 + bx + c) \, dx}$. Then the quadratic equation $ax^2 + bx + c = 0$ has:

If for a real number $y$,$[y]$ is the greatest integer less than or equal to $y$,then the value of the integral $\int_{\pi /2}^{3\pi /2} [2\sin x] \, dx$ is

Let $f(\alpha) = \int_{0}^{\alpha} x^{2} \left(1 - \frac{x}{\alpha}\right)^{\alpha} dx$ (where $\alpha > 0$),then $\sum_{\alpha=1}^{5} \frac{f(\alpha)}{\alpha^{3}}$ is equal to-

Let $I_n = \int_{0}^{\frac{\pi}{4}} \tan^n x \, dx$. Then $\frac{1}{I_2 + I_4}, \frac{1}{I_3 + I_5}, \frac{1}{I_4 + I_6}, \dots$ are in: