Dottie Parra

2021-11-08

Evaluate the integrals.
$\int \frac{\mathrm{cos}\left(\sqrt{x}\right)}{\sqrt{x}}dx$

Derrick

Step 1:To determine
To evaluate:
$\int \frac{\mathrm{cos}\left(\sqrt{x}\right)}{\sqrt{x}}dx$
Step 2:Calculation
Consider the given integral $\int \frac{\mathrm{cos}\left(\sqrt{x}\right)}{\sqrt{x}}dx$
Let $u=\sqrt{x}$
$⇒du=\frac{1}{2\sqrt{x}}dx$
$⇒2du=\frac{dx}{\sqrt{x}}$
So, the integral becomes,
$\int \frac{\mathrm{cos}\left(\sqrt{x}\right)}{\sqrt{x}}=\int \mathrm{cos}\left(u\right)2du$
$=2\int \mathrm{cos}\left(u\right)du$
$=2\mathrm{sin}\left(u\right)+C$
$=2\mathrm{sin}\left(\sqrt{x}\right)+C$ where C is the constant of integration.
Hence, $\int \frac{\mathrm{cos}\left(\sqrt{x}\right)}{\sqrt{x}}=2\mathrm{sin}\left(\sqrt{x}\right)+C$

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