integral of cos^3(y)sin^3(y)

解答

\int cos^{3} (y) sin^{3} (y) d y

- \frac{cos ^{4} ( y )}{4} + \frac{cos ^{6} ( y )}{6} + C

求解步骤

\int cos^{3} (y) sin^{3} (y) d y

使用三角恒等式改写

= \int (1 - cos^{2} (y)) sin (y) cos^{3} (y) d y

使用换元积分法

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

乘开

- u^{3} (1 - u^{2}) : - u^{3} + u^{5}

= \int - u^{3} + u^{5} d u

使用积分加法定则:

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

= - \int u^{3} d u + \int u^{5} d u

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

\int u^{5} d u = \frac{u ^{6}}{6}

= - \frac{u ^{4}}{4} + \frac{u ^{6}}{6}

u = cos (y)

代回

= - \frac{cos ^{4} ( y )}{4} + \frac{cos ^{6} ( y )}{6}

解答补常数

= - \frac{cos ^{4} ( y )}{4} + \frac{cos ^{6} ( y )}{6} + C