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

解答

\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

求解步骤

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

使用三角换元法

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

使用分布积分法

= 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)

u = sin (x)

代回

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

化简

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)

解答补常数

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

d er i v a t i v e f (x) = \frac{3 x + 2}{3 x - 2}

\frac{d}{d x} (3 x + 2 y)

\frac{d}{d x} (25 + x^{2})

s l o p e (2, - 2), (9, 3)

d er i v a t i v e 4 arcsin (x^{2})