What is the integral of (cos(sqrt(5x)))/(sqrt(5x)) ?

The integral of (cos(sqrt(5x)))/(sqrt(5x)) is 2/5 sin(sqrt(5)sqrt(x))+C

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integral of (cos(sqrt(5x)))/(sqrt(5x))

\int \frac{cos ( 5 x )}{5 x} d x

\frac{2}{5} sin (5 x) + C

Solution steps

\int \frac{cos ( 5 x )}{5 x} d x

Simplify

\frac{cos ( 5 x )}{5 x} : \frac{cos ( 5 x )}{5 x}

= \int \frac{cos ( 5 x )}{5 x} d x

Take the constant out:

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

= \frac{1}{5} \cdot \int \frac{cos ( 5 x )}{x} d x

Apply u-substitution

: \int \frac{2 cos ( u )}{5} d u

= \frac{1}{5} \cdot \int \frac{2 cos ( u )}{5} d u

Take the constant out:

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

= \frac{1}{5} \cdot \frac{2}{5} \cdot \int cos (u) d u

Use the common integral:

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

= \frac{1}{5} \cdot \frac{2}{5} sin (u)

Substitute back

u = 5 x

= \frac{1}{5} \cdot \frac{2}{5} sin (5 x)

Simplify

\frac{1}{5} \cdot \frac{2}{5} sin (5 x) : \frac{2}{5} sin (5 x)

= \frac{2}{5} sin (5 x)

Add a constant to the solution

= \frac{2}{5} sin (5 x) + C

\int x (x - 2)^{7} d x

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\int \frac{cos ( x )}{1 + tan ( x )} d x

What is the integral of (cos(sqrt(5x)))/(sqrt(5x)) ?
The integral of (cos(sqrt(5x)))/(sqrt(5x)) is 2/5 sin(sqrt(5)sqrt(x))+C