Solved: "Evaluate the integrals. ∫cos⁡(x)xdx"

Dottie Parra

Answered question

2021-11-08

Evaluate the integrals.

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

Derrick

Skilled2021-11-09Added 94 answers

Step 1:To determine
To evaluate:

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

Step 2:Calculation
Consider the given integral

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

Let

u = \sqrt{x}

\Rightarrow d u = \frac{1}{2 \sqrt{x}} d x

\Rightarrow 2 d u = \frac{d x}{\sqrt{x}}

So, the integral becomes,

\int \frac{\cos (\sqrt{x})}{\sqrt{x}} = \int \cos (u) 2 d u

= 2 \int \cos (u) d u

= 2 \sin (u) + C

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

where C is the constant of integration.
Hence,

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

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