What is the integral of (cos(sqrt(7x)))/(sqrt(7x)) ?

The integral of (cos(sqrt(7x)))/(sqrt(7x)) is 2/7 sin(sqrt(7)sqrt(x))+C

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integral of (cos(sqrt(7x)))/(sqrt(7x))

\int \frac{cos ( 7 x )}{7 x} d x

\frac{2}{7} sin (7 x) + C

Solution steps

\int \frac{cos ( 7 x )}{7 x} d x

Simplify

\frac{cos ( 7 x )}{7 x} : \frac{cos ( 7 x )}{7 x}

= \int \frac{cos ( 7 x )}{7 x} d x

Take the constant out:

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

= \frac{1}{7} \cdot \int \frac{cos ( 7 x )}{x} d x

Apply u-substitution

: \int \frac{2 cos ( u )}{7} d u

= \frac{1}{7} \cdot \int \frac{2 cos ( u )}{7} d u

Take the constant out:

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

= \frac{1}{7} \cdot \frac{2}{7} \cdot \int cos (u) d u

Use the common integral:

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

= \frac{1}{7} \cdot \frac{2}{7} sin (u)

Substitute back

u = 7 x

= \frac{1}{7} \cdot \frac{2}{7} sin (7 x)

Simplify

\frac{1}{7} \cdot \frac{2}{7} sin (7 x) : \frac{2}{7} sin (7 x)

= \frac{2}{7} sin (7 x)

Add a constant to the solution

= \frac{2}{7} sin (7 x) + C

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What is the integral of (cos(sqrt(7x)))/(sqrt(7x)) ?
The integral of (cos(sqrt(7x)))/(sqrt(7x)) is 2/7 sin(sqrt(7)sqrt(x))+C