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If $I_n = \int_{0}^{\pi} \frac{\sin(nx)}{\sin(x)} dx$,then the value of $\sum_{n=1}^{10} I_n$ is-

The value of the integral $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left(x^2 + \log \frac{\pi-x}{\pi+x}\right) \cos x \, dx$ is equal to

$\frac{3}{25} \int_0^{25 \pi} \sqrt{|\cos x - \cos^3 x|} \, dx =$

$\int_{-1}^{1} x|x| \, dx = $

If $I_n = \int_0^{\pi / 4} \tan^n x \, dx$,then $\frac{1}{I_2 + I_4} + \frac{1}{I_3 + I_5} + \frac{1}{I_4 + I_6} = $

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