Mathematics

The value of $$\displaystyle\int { \cfrac { \cos { \sqrt { x }  }  }{ \sqrt { x }  }  } dx$$ is


ANSWER

$$2\sin { \sqrt { x } } +C$$


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

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

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

Now,$$\displaystyle\int{\dfrac{\cos{\sqrt{x}}}{\sqrt{x}}dx}$$

$$=\displaystyle\int{2dt}=2t+c$$ ......where $$c$$ is the constant of integration

$$=2\sin{\sqrt{x}}+c$$ ...........where $$t=\sin{\sqrt{x}}$$
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Single Correct Hard Published on 17th 09, 2020
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