gorovogpg

2022-01-01

Find integral.
$\int \frac{\mathrm{cos}x}{\sqrt{\mathrm{sin}x}}dx$

Joseph Fair

Step 1:To determine
Evaluate:
$\int \frac{\mathrm{cos}x}{\sqrt{\mathrm{sin}x}}dx$
Step 2:Formula used
$\int {x}^{n}dx=\frac{{x}^{n+1}}{n+1}$
$\frac{d}{dx}\left(\mathrm{sin}x\right)=\mathrm{cos}x$
Step 3:Solution
Consider the given integral:
$\int \frac{\mathrm{cos}x}{\sqrt{\mathrm{sin}x}}dx$
Let $\mathrm{sin}x=t⇒\mathrm{cos}xdx=dt$
By subsitution, the integral becomes:
$\int \frac{dt}{\sqrt{t}}$
$=\int {t}^{\frac{-1}{2}}dt$
$=\frac{{t}^{\frac{-1}{2}+1}}{\frac{-1}{2}+1}+C$. where C is the constant of integration.
$=\frac{{t}^{\frac{1}{2}}}{\frac{1}{2}}+C$
$=2\sqrt{t}+C$
$=2\sqrt{\mathrm{sin}x}+C$
Hence, $\int \frac{\mathrm{cos}x}{\sqrt{\mathrm{sin}x}}dx=2\sqrt{\mathrm{sin}x}+C$
Step 4:Conclusion
Hence, $\int \frac{\mathrm{cos}x}{\sqrt{\mathrm{sin}x}}dx=2\sqrt{\mathrm{sin}x}+C$

Robert Pina

Given:
$\int \frac{\mathrm{cos}\left(x\right)}{\sqrt{\mathrm{sin}\left(x\right)}}dx$
$\int \frac{\mathrm{cos}\left(x\right)}{{\mathrm{sin}\left(x\right)}^{\frac{1}{2}}}dx$
$\int \frac{1}{{t}^{\frac{1}{2}}}dt$
Evaluate
$2\sqrt{t}$
$2\sqrt{\mathrm{sin}\left(x\right)}$
$2\sqrt{\mathrm{sin}\left(x\right)}+C$

Vasquez

$\int \frac{\mathrm{cos}\left(x\right)}{\sqrt{\mathrm{sin}\left(x\right)}}dx$
We put the expression $\mathrm{cos}\left(x\right)$ under the differential sign, i.e.:
$\mathrm{cos}\left(x\right)dx=d\left(\mathrm{sin}\left(x\right)\right),t=\mathrm{sin}\left(x\right)$
Then the original integral can be written as follows:
$\int \frac{1}{\sqrt{t}}dt$
This is a tabular integral:
$\int \frac{1}{\sqrt{t}}dt=2\sqrt{t}+C$
To write the final The answer is, it remains to substitute $\mathrm{sin}\left(x\right)$ instead of t.
$2\sqrt{\mathrm{sin}\left(x\right)}+C$

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