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

The integral of sin(ln(x^3)) is -3/10 xcos(3ln(x))+1/10 xsin(3ln(x))+C

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integral of sin(ln(x^3))

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

- \frac{3}{10} x cos (3 ln (x)) + \frac{1}{10} x sin (3 ln (x)) + C

Solution steps

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

Simplify

ln (x^{3}) : 3 ln (x)

= \int sin (3 ln (x)) d x

Apply u-substitution

: \int sin (3 u) e^{u} d u

= \int sin (3 u) e^{u} d u

Apply Integration By Parts

: - \frac{1}{3} e^{u} cos (3 u) - \int - \frac{1}{3} e^{u} cos (3 u) d u

= - \frac{1}{3} e^{u} cos (3 u) - \int - \frac{1}{3} e^{u} cos (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} e^{u} cos (3 u) - (- \frac{1}{3} \cdot \int e^{u} cos (3 u) d u)

Apply Integration By Parts

: e^{u} \frac{1}{3} sin (3 u) - \int e^{u} \frac{1}{3} sin (3 u) d u

= - \frac{1}{3} e^{u} cos (3 u) - (- \frac{1}{3} (e^{u} \frac{1}{3} sin (3 u) - \int e^{u} \frac{1}{3} sin (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} e^{u} cos (3 u) - (- \frac{1}{3} (e^{u} \frac{1}{3} sin (3 u) - \frac{1}{3} \cdot \int e^{u} sin (3 u) d u))

Therefore

\int e^{u} sin (3 u) d u = - \frac{1}{3} e^{u} cos (3 u) - (- \frac{1}{3} (e^{u} \frac{1}{3} sin (3 u) - \frac{1}{3} \cdot \int e^{u} sin (3 u) d u))

Isolate

\int e^{u} sin (3 u) d u

= - \frac{3 e ^{u} cos ( 3 u )}{10} + \frac{e ^{u} sin ( 3 u )}{10}

Substitute back

u = ln (x)

= - \frac{3 e ^{l n (x)} cos ( 3 ln ( x ) )}{10} + \frac{e ^{l n (x)} sin ( 3 ln ( x ) )}{10}

Simplify

- \frac{3 e ^{l n (x)} cos ( 3 ln ( x ) )}{10} + \frac{e ^{l n (x)} sin ( 3 ln ( x ) )}{10} : - \frac{3}{10} x cos (3 ln (x)) + \frac{1}{10} x sin (3 ln (x))

= - \frac{3}{10} x cos (3 ln (x)) + \frac{1}{10} x sin (3 ln (x))

Add a constant to the solution

= - \frac{3}{10} x cos (3 ln (x)) + \frac{1}{10} x sin (3 ln (x)) + C

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What is the integral of sin(ln(x^3)) ?
The integral of sin(ln(x^3)) is -3/10 xcos(3ln(x))+1/10 xsin(3ln(x))+C