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

The integral of cos^3(x)ln(sin(x)) is -1/3 sin^3(x)ln(sin(x))+ln(sin(x))sin(x)+1/9 sin^3(x)-sin(x)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

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

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

- \frac{1}{3} sin^{3} (x) ln (sin (x)) + ln (sin (x)) sin (x) + \frac{1}{9} sin^{3} (x) - sin (x) + C

Solution steps

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

Apply Trigonometric Substitution

= \int ln (u) (- u^{2} + 1) d u

Apply Integration By Parts

= ln (u) (- \frac{u ^{3}}{3} + u) - \int \frac{1}{3} (- u^{2} + 3) d u

\int \frac{1}{3} (- u^{2} + 3) d u = - \frac{u ^{3}}{9} + u

= ln (u) (- \frac{u ^{3}}{3} + u) - (- \frac{u ^{3}}{9} + u)

Substitute back

u = sin (x)

= ln (sin (x)) (- \frac{sin ^{3} ( x )}{3} + sin (x)) - (- \frac{sin ^{3} ( x )}{9} + sin (x))

Simplify

ln (sin (x)) (- \frac{sin ^{3} ( x )}{3} + sin (x)) - (- \frac{sin ^{3} ( x )}{9} + sin (x)) : - \frac{1}{3} sin^{3} (x) ln (sin (x)) + ln (sin (x)) sin (x) + \frac{1}{9} sin^{3} (x) - sin (x)

= - \frac{1}{3} sin^{3} (x) ln (sin (x)) + ln (sin (x)) sin (x) + \frac{1}{9} sin^{3} (x) - sin (x)

Add a constant to the solution

= - \frac{1}{3} sin^{3} (x) ln (sin (x)) + ln (sin (x)) sin (x) + \frac{1}{9} sin^{3} (x) - sin (x) + C

What is the integral of cos^3(x)ln(sin(x)) ?
The integral of cos^3(x)ln(sin(x)) is -1/3 sin^3(x)ln(sin(x))+ln(sin(x))sin(x)+1/9 sin^3(x)-sin(x)+C