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受欢迎的微积分 >

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

解答

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

解答

\frac{1}{4} (\frac{sin ^{5} ( 4 x )}{5} - \frac{3 sin ^{7} ( 4 x )}{7} + \frac{sin ^{9} ( 4 x )}{3} - \frac{sin ^{11} ( 4 x )}{11}) + C

求解步骤

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

使用换元积分法

= \int cos^{7} (u) sin^{4} (u) \frac{1}{4} d u

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{4} \cdot \int cos^{7} (u) sin^{4} (u) d u

使用三角恒等式改写

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

使用换元积分法

= \frac{1}{4} \cdot \int v^{4} (1 - v^{2})^{3} d v

乘开

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

= \frac{1}{4} \cdot \int v^{4} - 3 v^{6} + 3 v^{8} - v^{10} d v

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

= \frac{1}{4} (\int v^{4} d v - \int 3 v^{6} d v + \int 3 v^{8} d v - \int v^{10} d v)

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

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

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

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

= \frac{1}{4} (\frac{v ^{5}}{5} - \frac{3 v ^{7}}{7} + \frac{v ^{9}}{3} - \frac{v ^{11}}{11})

代回

= \frac{1}{4} (\frac{sin ^{5} ( 4 x )}{5} - \frac{3 sin ^{7} ( 4 x )}{7} + \frac{sin ^{9} ( 4 x )}{3} - \frac{sin ^{11} ( 4 x )}{11})

解答补常数

= \frac{1}{4} (\frac{sin ^{5} ( 4 x )}{5} - \frac{3 sin ^{7} ( 4 x )}{7} + \frac{sin ^{9} ( 4 x )}{3} - \frac{sin ^{11} ( 4 x )}{11}) + C

作图

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