integral of sin^4(x)cos^5(x)

解答

\int sin^{4} (x) cos^{5} (x) d x

\frac{sin ^{5} ( x )}{5} - \frac{2 sin ^{7} ( x )}{7} + \frac{sin ^{9} ( x )}{9} + C

求解步骤

\int sin^{4} (x) cos^{5} (x) d x

使用三角恒等式改写

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

使用换元积分法

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

乘开

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

= \int u^{4} - 2 u^{6} + u^{8} 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 2 u^{6} d u + \int u^{8} d u

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

\int 2 u^{6} d u = \frac{2 u ^{7}}{7}

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

= \frac{u ^{5}}{5} - \frac{2 u ^{7}}{7} + \frac{u ^{9}}{9}

u = sin (x)

代回

= \frac{sin ^{5} ( x )}{5} - \frac{2 sin ^{7} ( x )}{7} + \frac{sin ^{9} ( x )}{9}

解答补常数

= \frac{sin ^{5} ( x )}{5} - \frac{2 sin ^{7} ( x )}{7} + \frac{sin ^{9} ( x )}{9} + C

d er i v a t i v e f (x) = 7 e^{x} + \frac{4}{3 x}

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