What is the integral of (sin(ln(x^3)))/x ?

The integral of (sin(ln(x^3)))/x is -1/3 cos(3ln(x))+C

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integral of (sin(ln(x^3)))/x

\int \frac{sin ( ln ( x ^{3} ) )}{x} d x

- \frac{1}{3} cos (3 ln (x)) + C

Solution steps

\int \frac{sin ( ln ( x ^{3} ) )}{x} d x

Apply u-substitution

: \int \frac{sin ( ln ( u ) )}{3 u} d u

= \int \frac{sin ( ln ( u ) )}{3 u} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int \frac{sin ( ln ( u ) )}{u} d u

Apply u-substitution

: \int sin (v) d v

= \frac{1}{3} \cdot \int sin (v) d v

Use the common integral:

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

= \frac{1}{3} (- cos (v))

Substitute back

= \frac{1}{3} (- cos (ln (x^{3})))

Simplify

\frac{1}{3} (- cos (ln (x^{3}))) : - \frac{1}{3} cos (3 ln (x))

= - \frac{1}{3} cos (3 ln (x))

Add a constant to the solution

= - \frac{1}{3} cos (3 ln (x)) + C

a e^{3 x}

g (x) = (x + 2 x) e^{x}

\frac{1}{( s - 2 ) ^{2}}

\frac{d ^{3}}{d x ^{3}} (\frac{1}{x ^{2} + 1})

8 x^{2} + 3 x - 7

What is the integral of (sin(ln(x^3)))/x ?
The integral of (sin(ln(x^3)))/x is -1/3 cos(3ln(x))+C