Mathematics

Write a value of 
$$\displaystyle\int { \cfrac { \cos { x }  }{ \sin { x } \log { \sin { x }  }  }  } dx\quad \quad $$


SOLUTION
Let $$t=\sin{x}\Rightarrow\,dt=\cos{x}dx$$

$$\displaystyle\int{\dfrac{\cos{x}dx}{\sin{x}\log{\sin{x}}}}$$

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

Let $$u=\log{t}\Rightarrow\,du=\dfrac{1}{t}dt$$

$$=\displaystyle\int{\dfrac{du}{u}}$$

$$=\log{\left(u\right)}+c$$     ..........where $$c$$ is the constant of integration.

$$=\log{\left(\log{t}\right)}+c$$     ............where $$u=\log{t}$$

$$=\log{\left(\log{\sin{x}}\right)}+c$$      ..........where $$t=\sin{x}$$
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Subjective Medium Published on 17th 09, 2020
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