Mathematics

Evaluate the following integral:
$$\int { \cfrac { 1+\cos { x }  }{ { \left( x+\sin { x }  \right)  }^{ 3 } }  } dx\quad $$


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

$$\displaystyle\int{\dfrac{\left(1+\cos{x}\right)dx}{{\left(x+\sin{x}\right)}^{3}}}$$

$$=\displaystyle\int{\dfrac{dt}{{t}^{3}}}$$

$$=\displaystyle\int{{t}^{-3}dt}$$

$$=\dfrac{{t}^{-3+1}}{-3+1}+c$$

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

$$=\dfrac{-1}{2{\left(x+\sin{x}\right)}^{2}}+c$$ where $$t=x+\sin{x}$$
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Subjective Medium Published on 17th 09, 2020
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