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

The integral of sin^3(x)cos(x)ln(sin(x)) is 1/4 sin^4(x)ln(sin(x))-(sin^4(x))/(16)+C

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

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

\frac{1}{4} sin^{4} (x) ln (sin (x)) - \frac{sin ^{4} ( x )}{16} + C

Solution steps

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

Apply u-substitution

: \int u^{3} ln (u) d u

= \int u^{3} ln (u) d u

Apply Integration By Parts

: \frac{1}{4} u^{4} ln (u) - \int \frac{u ^{3}}{4} d u

= \frac{1}{4} u^{4} ln (u) - \int \frac{u ^{3}}{4} d u

\int \frac{u ^{3}}{4} d u = \frac{u ^{4}}{16}

= \frac{1}{4} u^{4} ln (u) - \frac{u ^{4}}{16}

Substitute back

u = sin (x)

= \frac{1}{4} sin^{4} (x) ln (sin (x)) - \frac{sin ^{4} ( x )}{16}

Add a constant to the solution

= \frac{1}{4} sin^{4} (x) ln (sin (x)) - \frac{sin ^{4} ( x )}{16} + C

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\int \frac{3 x ^{2} + 3 x - 23}{x ^{2} - 9} d x

What is the integral of sin^3(x)cos(x)ln(sin(x)) ?
The integral of sin^3(x)cos(x)ln(sin(x)) is 1/4 sin^4(x)ln(sin(x))-(sin^4(x))/(16)+C