7.2 definite integral — Maths STD 12 Science — Question

$I = \int\limits_{\frac{{3\pi }}{4}}^{\frac{{3\pi }}{4}} {\frac{{{\rm{dx}}}}{{1 - \cos x}}} $

Using$\int\limits_a^b {f\left( x \right)dx = \int\limits_a^b {f\left( {a + b - x} \right)} } dx$

Adding $(i)$ and $(ii)$

$2I = \int\limits_{\frac{\pi }{4}}^{\frac{{3\pi }}{4}} {\frac{2}{{{{\sin }^2}x}}dx} $

$I = \int\limits_{\frac{\pi }{4}}^{\frac{{3\pi }}{4}} {{{\csc }^2}xdx} $

${\rm{I}} =  - (\cot x)_{\pi /4}^{3\pi /4}$

$ =  - \left[ {\cot \frac{{3\pi }}{4} - \cot \frac{\pi }{4}} \right] = 2$