What is the integral of (sin(sqrt(x)))/(sqrt(4x)) ?

The integral of (sin(sqrt(x)))/(sqrt(4x)) is -cos(sqrt(x))+C

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integral of (sin(sqrt(x)))/(sqrt(4x))

\int \frac{sin ( x )}{4 x} d x

- cos (x) + C

Solution steps

\int \frac{sin ( x )}{4 x} d x

4 x = 2 x

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

Apply u-substitution

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

Use the common integral:

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

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

Substitute back

u = x

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

Simplify

\frac{1}{2} \cdot 2 (- cos (x)) : - cos (x)

= - cos (x)

Add a constant to the solution

= - cos (x) + C

What is the integral of (sin(sqrt(x)))/(sqrt(4x)) ?
The integral of (sin(sqrt(x)))/(sqrt(4x)) is -cos(sqrt(x))+C