What is the integral of (cos(sqrt(x)))/(2sqrt(x)) ?

The integral of (cos(sqrt(x)))/(2sqrt(x)) is sin(sqrt(x))+C

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integral of (cos(sqrt(x)))/(2sqrt(x))

\int \frac{cos ( x )}{2 x} d x

sin (x) + C

Solution steps

\int \frac{cos ( x )}{2 x} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int \frac{cos ( x )}{x} d x

Apply u-substitution

= \frac{1}{2} \cdot \int 2 cos (u) d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot 2 \cdot \int cos (u) d u

Use the common integral:

\int cos (u) d u = sin (u)

= \frac{1}{2} \cdot 2 sin (u)

Substitute back

u = x

= \frac{1}{2} \cdot 2 sin (x)

Simplify

\frac{1}{2} \cdot 2 sin (x) : sin (x)

= sin (x)

Add a constant to the solution

= sin (x) + C

What is the integral of (cos(sqrt(x)))/(2sqrt(x)) ?
The integral of (cos(sqrt(x)))/(2sqrt(x)) is sin(sqrt(x))+C