integral of cos^1(x)sin^2(x)

解答

\int cos^{1} (x) sin^{2} (x) d x

\frac{sin ^{3} ( x )}{3} + C

求解步骤

\int cos^{1} (x) sin^{2} (x) d x

化简

sin^{2} (x) : sin (x) sin (x)

= \int cos^{1} (x) sin (x) sin (x) d x

化简

cos^{1} (x) sin (x) sin (x) : sin^{2} (x) cos (x)

= \int sin^{2} (x) cos (x) d x

使用换元积分法

= \int u^{2} d u

使用幂法则

= \frac{u ^{3}}{3}

u = sin (x)

代回

= \frac{sin ^{3} ( x )}{3}

解答补常数

= \frac{sin ^{3} ( x )}{3} + C

\int \frac{x ^{2} + 1}{( x ^{2} - 1 ) ( x + 1 )} d x

\int \frac{ln ^{4} ( x )}{x ( ln ^{2} ( x ) - 1 )} d x

\int \frac{sin ( 2 x )}{30 + cos ^{2} ( x )} d x

\int cos (4 x) cos (x) d x

\int (x - \frac{1}{x}) d x