Mathematics

The integral $$\int {\sqrt {\cot x} \,{e^{\sqrt {\sin x} }}\sqrt {\cos x} \,dx} $$ equals


SOLUTION
$$\displaystyle\int{\sqrt{\cot{x}}{e}^{\sqrt{\sin{x}}}\sqrt{\cos{x}}\,dx}$$

$$=\displaystyle\int{\sqrt{\cot{x}\cos{x}}{e}^{\sqrt{\sin{x}}}\,dx}$$

$$=\displaystyle\int{\sqrt{\dfrac{\cos{x}}{\sin{x}}\cos{x}}{e}^{\sqrt{\sin{x}}}\,dx}$$

$$=\displaystyle\int{\dfrac{\cos{x}}{\sqrt{\sin{x}}}e^{\sqrt{\sin{x}}}dx}$$

Let $$t=\sqrt{\sin{x}}\Rightarrow\,dt=\dfrac{\cos{x}}{2\sqrt{\sin{x}}}dx$$

$$=\displaystyle\int{2{e}^{t}dt}$$

$$=2\displaystyle\int{{e}^{t}dt}$$

$$=2{e}^{t}+c$$

$$=2{e}^{\sqrt{\sin{x}}}+c$$ where $$t=\sqrt{\sin{x}}$$
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Subjective Medium Published on 17th 09, 2020
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