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

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

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

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

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

Solution steps

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

Simplify

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

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

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 t )}{t} d t

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 t

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

Simplify

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

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

Add a constant to the solution

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

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