Mathematics

Evaluate the following integral:
$$\int { \cfrac { 1+\sin { x }  }{ \sqrt { x-\cos { x }  }  }  } dx$$


SOLUTION
Let 
$$t=x-\cos{x}\Rightarrow\,dt=\left(1+\sin{x}\right)dx$$

$$\displaystyle\int{\dfrac{\left(1+\sin{x}\right)dx}{\sqrt{x-\cos{x}}}}$$

$$=\displaystyle\int{\dfrac{dt}{\sqrt{t}}}$$

$$=\displaystyle\int{{t}^{\frac{-1}{2}}dt}$$

$$=\dfrac{{t}^{\frac{-1}{2}+1}}{\dfrac{-1}{2}+1}+c$$

$$=\dfrac{{t}^{\frac{1}{2}}}{\dfrac{1}{2}}+c$$

$$=2\sqrt{t}+c$$

$$=2\sqrt{x-\cos{x}}+c$$ where $$t=x-\cos{x}$$

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Subjective Medium Published on 17th 09, 2020
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