What is the integral of (cos(ln(x^2)))/(5x) ?

The integral of (cos(ln(x^2)))/(5x) is 1/10 sin(2ln(x))+C

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integral of (cos(ln(x^2)))/(5x)

\int \frac{cos ( ln ( x ^{2} ) )}{5 x} d x

\frac{1}{10} sin (2 ln (x)) + C

Solution steps

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

\frac{cos ( ln ( x ^{2} ) )}{x} = cos (ln (x^{2})) \frac{1}{x}

= \frac{1}{5} \cdot \int cos (ln (x^{2})) \frac{1}{x} d x

Apply u-substitution

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

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

Apply u-substitution

: \int cos (v) d v

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

Use the common integral:

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

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

Substitute back

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

Simplify

\frac{1}{5} \cdot \frac{1}{2} sin (ln (x^{2})) : \frac{1}{10} sin (2 ln (x))

= \frac{1}{10} sin (2 ln (x))

Add a constant to the solution

= \frac{1}{10} sin (2 ln (x)) + C

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What is the integral of (cos(ln(x^2)))/(5x) ?
The integral of (cos(ln(x^2)))/(5x) is 1/10 sin(2ln(x))+C