Answered: "Find integral. ∫cos⁡xsin⁡xdx"

High School
Calculus and Analysis
Calculus 2
Applications of integrals

gorovogpg

Answered question

2022-01-01

Find integral.

\int \frac{\cos x}{\sqrt{\sin x}} d x

Answer & Explanation

Joseph Fair

Beginner2022-01-02Added 34 answers

Step 1:To determine
Evaluate:

\int \frac{\cos x}{\sqrt{\sin x}} d x

Step 2:Formula used

\int x^{n} d x = \frac{x^{n + 1}}{n + 1}

\frac{d}{d x} (\sin x) = \cos x

Step 3:Solution
Consider the given integral:

\int \frac{\cos x}{\sqrt{\sin x}} d x

Let

\sin x = t \Rightarrow \cos x d x = d t

By subsitution, the integral becomes:

\int \frac{d t}{\sqrt{t}}

= \int t^{\frac{- 1}{2}} d t

= \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{\sin x} + C

Hence,

\int \frac{\cos x}{\sqrt{\sin x}} d x = 2 \sqrt{\sin x} + C

Step 4:Conclusion
Hence,

\int \frac{\cos x}{\sqrt{\sin x}} d x = 2 \sqrt{\sin x} + C

Robert Pina

Beginner2022-01-03Added 42 answers

Given:

\int \frac{\cos (x)}{\sqrt{\sin (x)}} d x

\int \frac{\cos (x)}{{\sin (x)}^{\frac{1}{2}}} d x

\int \frac{1}{t^{\frac{1}{2}}} d t

Evaluate

2 \sqrt{t}

2 \sqrt{\sin (x)}

Answer:

2 \sqrt{\sin (x)} + C

Vasquez

Expert2022-01-07Added 669 answers

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

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