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
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