integral of sin^4(y)cos^7(y)

解答

\int sin^{4} (y) cos^{7} (y) d y

\frac{sin ^{5} ( y )}{5} - \frac{3 sin ^{7} ( y )}{7} + \frac{sin ^{9} ( y )}{3} - \frac{sin ^{11} ( y )}{11} + C

求解步骤

\int sin^{4} (y) cos^{7} (y) d y

使用三角恒等式改写

= \int (1 - sin^{2} (y))^{3} cos (y) sin^{4} (y) d y

使用换元积分法

= \int u^{4} (1 - u^{2})^{3} d u

乘开

u^{4} (1 - u^{2})^{3} : u^{4} - 3 u^{6} + 3 u^{8} - u^{10}

= \int u^{4} - 3 u^{6} + 3 u^{8} - u^{10} 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^{4} d u - \int 3 u^{6} d u + \int 3 u^{8} d u - \int u^{10} d u

\int u^{4} d u = \frac{u ^{5}}{5}

\int 3 u^{6} d u = \frac{3 u ^{7}}{7}

\int 3 u^{8} d u = \frac{u ^{9}}{3}

\int u^{10} d u = \frac{u ^{11}}{11}

= \frac{u ^{5}}{5} - \frac{3 u ^{7}}{7} + \frac{u ^{9}}{3} - \frac{u ^{11}}{11}

u = sin (y)

代回

= \frac{sin ^{5} ( y )}{5} - \frac{3 sin ^{7} ( y )}{7} + \frac{sin ^{9} ( y )}{3} - \frac{sin ^{11} ( y )}{11}

解答补常数

= \frac{sin ^{5} ( y )}{5} - \frac{3 sin ^{7} ( y )}{7} + \frac{sin ^{9} ( y )}{3} - \frac{sin ^{11} ( y )}{11} + C

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