Mathematics

State whether the statement is ture/false.

 $$\displaystyle \int _ { - \pi / 2 } ^ { \pi / 2 } \left( \frac { \sin x } { 1 - \cos x } \right) d x$$=0


ANSWER

True


SOLUTION
$$\displaystyle\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}{\dfrac{\sin{x}}{1-\cos{x}}dx}$$
$$=\displaystyle\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}{\dfrac{2\sin{\dfrac{x}{2}}\cos{\dfrac{x}{2}}}{2{\sin}^{2}{\dfrac{x}{2}}}dx}$$
$$=\displaystyle\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}{\dfrac{\cos{\dfrac{x}{2}}}{\sin{\dfrac{x}{2}}}dx}$$
$$=\displaystyle\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}{\cot{\dfrac{x}{2}}dx}$$
$$=2\left[\ln{\left|\sin{\dfrac{x}{2}}\right|}\right]_{-\frac{\pi}{2}}^{\frac{\pi}{2}}$$
$$=2\left[\ln{\left|\sin{\dfrac{\pi}{4}}\right|}-\ln{\left|\sin{\dfrac{\pi}{4}}\right|}\right]=0$$
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TRUE/FALSE Medium Published on 17th 09, 2020
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