Value of the $\int_0^9 \sqrt{x} \,dx + \int_0^{\pi/2} (\cos x + \sin x) \,dx$ is:

If $I = \sum_{k=1}^{98} \int_k^{k+1} \frac{k+1}{x(x+1)} dx$,then which of the following is true?

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

Let $I_1 = \int_0^{\pi/2} \frac{\sin x - \cos x}{1 + \sin x \cos x} dx$,$I_2 = \int_0^{2\pi} \cos^6 x dx$,$I_3 = \int_{-\pi/2}^{\pi/2} \sin^3 x dx$,and $I_4 = \int_0^1 \ln \left( \frac{1}{x} - 1 \right) dx$. Then:

The number of continuous functions $f : [0, \frac{3}{2}] \rightarrow (0, \infty)$ satisfying the equation $4 \int_0^{3/2} f(x) dx + 125 \int_0^{3/2} \frac{dx}{\sqrt{f(x)+x^2}} = 108$ is

If $m \in Z^{+}$,$n=2m$ and $\int_0^{\frac{\pi}{2}} \sin ^{m} x \cos ^{n} x \, dx = K(m) \int_0^{\frac{\pi}{2}} \sin ^m x \, dx$,then $\frac{2^{m-1}(m-1)!}{(2m-1)!} K(m) =$