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

The integral of (sin^5(ln(x)))/x is -cos(ln(x))+(2cos^3(ln(x)))/3-(cos^5(ln(x)))/5+C

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integral of (sin^5(ln(x)))/x

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

- cos (ln (x)) + \frac{2 cos ^{3} ( ln ( x ) )}{3} - \frac{cos ^{5} ( ln ( x ) )}{5} + C

Solution steps

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

Apply u-substitution

: \int sin^{5} (u) d u

= \int sin^{5} (u) d u

Rewrite using trig identities

= \int (1 - cos^{2} (u))^{2} sin (u) d u

Apply u-substitution

: \int - 1 + 2 v^{2} - v^{4} d v

= \int - 1 + 2 v^{2} - v^{4} d v

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \int 1 d v + \int 2 v^{2} d v - \int v^{4} d v

\int 1 d v = v

\int 2 v^{2} d v = \frac{2 v ^{3}}{3}

\int v^{4} d v = \frac{v ^{5}}{5}

= - v + \frac{2 v ^{3}}{3} - \frac{v ^{5}}{5}

Substitute back

= - cos (ln (x)) + \frac{2 cos ^{3} ( ln ( x ) )}{3} - \frac{cos ^{5} ( ln ( x ) )}{5}

Add a constant to the solution

= - cos (ln (x)) + \frac{2 cos ^{3} ( ln ( x ) )}{3} - \frac{cos ^{5} ( ln ( x ) )}{5} + C

x \to 0 lim (\frac{5 + x - 5}{2 x})

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f (x) = x (27 x^{- 2} + 2), at x = - 3

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What is the integral of (sin^5(ln(x)))/x ?
The integral of (sin^5(ln(x)))/x is -cos(ln(x))+(2cos^3(ln(x)))/3-(cos^5(ln(x)))/5+C