Mathematics

# Evaluate the given integral.$\displaystyle \int { \cfrac { \sin { \sqrt { x } } }{ \sqrt { x } } } dx$

##### SOLUTION
Let $t=\sqrt{x}$

$\Rightarrow\,dt=\dfrac{1}{2}{x}^{\frac{1}{2}-1}dx=\dfrac{1}{2}{x}^{\frac{-1}{2}}dx=\dfrac{1}{2\sqrt{x}}dx$

$I=\displaystyle\int{\dfrac{\sin{\sqrt{x}}}{\sqrt{x}}dx}$

$=2\displaystyle\int{\sin{t}dt}$

Let $u=\cos{t}\Rightarrow\,du=-\sin{t}dt$

$=-2\displaystyle\int{du}$

$=-2u+c$ ..........where $u=\cos{t}$

$=2{\cos}^{-1}{t}+c$ ............where $c$ is the constant of integration.

$=2{\cos}^{-1}{\sqrt{x}}+c$ ..........where $t=\sqrt{x}$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### Realted Questions

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