In $I(m, n) = \int_0^1 x^{m-1}(1-x)^{n-1} dx$,where $m, n > 0$,then $I(9, 14) + I(10, 13)$ is equal to:

Let ${I_1} = \int\limits_0^1 {\frac{{{e^x}}}{{1 + x}}} \,dx$ and ${I_2} = \int\limits_0^1 {\frac{{{x^2}}}{{{e^{{x^3}}}\left( {2 - {x^3}} \right)}}} \,dx$,then the value of $\frac{{{I_1}}}{{{I_2}}}$ is equal to

$ 6\int_{0}^{\pi}|(\sin 3x+\sin 2x+\sin x)| dx $ is equal to ....

Let $f(x) = \min \{[x-1], [x-2], \ldots, [x-10]\}$ where $[t]$ denotes the greatest integer $\leq t$. Then $\int_{0}^{10} f(x) \, dx + \int_{0}^{10} (f(x))^2 \, dx + \int_{0}^{10} |f(x)| \, dx$ is equal to

If $f(x) = \text{Max}\{\sin x, \cos x\}$ and $g(x) = \text{Min}\{\sin x, \cos x\}$,then $\int_{0}^{\pi} f(x) dx + \int_{0}^{\pi} g(x) dx = $

Let $I_{n} = \int_{0}^{1} x^{n} \tan^{-1} x \, dx$. If $a_{n} I_{n+2} + b_{n} I_{n} = c_{n}$ for all $n \geq 1$, then