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
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

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