Mathematics

$$\int (1- \cos x)\ \text{cosec}^2x \;dx  = f(x) +c\; implies\; f(x) = $$


ANSWER

$$\tan \dfrac{x}{2}$$+C


SOLUTION
$$\int { \left( 1-\cos x \right) { \cos ec }^{ 2 }xdx } $$
$$\Rightarrow \int { { \cos ec }^{ 2 }xdx-\int { \dfrac { \cos x }{ { \sin  }^{ 2 }x } dx }  } $$
$$\sin x=m$$
$$\cos xdx=dm$$
$$\Rightarrow -cotx-\int { \dfrac { dm }{ { m }^{ 2 } }  } $$
$$\Rightarrow -cotx+\dfrac { 1 }{ m } $$
$$\Rightarrow -cotx+\dfrac { 1 }{ \sin x } $$
$$\Rightarrow \dfrac { 1-\cos x }{ \sin x } $$
$$\Rightarrow \dfrac { 2{ \sin  }^{ 2 }\left( x/2 \right)  }{ 2\sin \left( \dfrac { x }{ 2 }  \right) \cos \left( \dfrac { x }{ 2 }  \right)  } =\tan\left( x/2 \right) +c$$
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Single Correct Medium Published on 17th 09, 2020
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