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受欢迎的微积分 >

integral of sin^4(5x)cos^6(5x)

解答

\int sin^{4} (5 x) cos^{6} (5 x) d x

解答

\frac{1}{12800} (150 x + 16 cos^{5} (5 x) sin (5 x) + 20 cos^{3} (5 x) sin (5 x) + 15 sin (10 x) - 352 cos^{7} (5 x) sin (5 x) + 256 cos^{9} (5 x) sin (5 x)) + C

求解步骤

\int sin^{4} (5 x) cos^{6} (5 x) d x

使用换元积分法

= \int sin^{4} (u) cos^{6} (u) \frac{1}{5} d u

提出常数:

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

= \frac{1}{5} \cdot \int sin^{4} (u) cos^{6} (u) d u

使用三角恒等式改写

= \frac{1}{5} \cdot \int (1 - cos^{2} (u))^{2} cos^{6} (u) d u

乘开

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

= \frac{1}{5} \cdot \int cos^{6} (u) - 2 cos^{8} (u) + cos^{10} (u) 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

= \frac{1}{5} (\int cos^{6} (u) d u - \int 2 cos^{8} (u) d u + \int cos^{10} (u) d u)

\int cos^{6} (u) d u = \frac{sin ( u ) cos ^{5} ( u )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (u) sin (u) + \frac{3}{8} (u + \frac{1}{2} sin (2 u)))

\int 2 cos^{8} (u) d u = 2 (\frac{sin ( u ) cos ^{7} ( u )}{8} + \frac{7}{8} (\frac{sin ( u ) cos ^{5} ( u )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (u) sin (u) + \frac{3}{8} (u + \frac{1}{2} sin (2 u)))))

\int cos^{10} (u) d u = \frac{sin ( u ) cos ^{9} ( u )}{10} + \frac{9}{10} (\frac{sin ( u ) cos ^{7} ( u )}{8} + \frac{7}{8} (\frac{sin ( u ) cos ^{5} ( u )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (u) sin (u) + \frac{3}{8} (u + \frac{1}{2} sin (2 u)))))

= \frac{1}{5} (\frac{sin ( u ) cos ^{5} ( u )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (u) sin (u) + \frac{3}{8} (u + \frac{1}{2} sin (2 u))) - 2 (\frac{sin ( u ) cos ^{7} ( u )}{8} + \frac{7}{8} (\frac{sin ( u ) cos ^{5} ( u )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (u) sin (u) + \frac{3}{8} (u + \frac{1}{2} sin (2 u))))) + \frac{sin ( u ) cos ^{9} ( u )}{10} + \frac{9}{10} (\frac{sin ( u ) cos ^{7} ( u )}{8} + \frac{7}{8} (\frac{sin ( u ) cos ^{5} ( u )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (u) sin (u) + \frac{3}{8} (u + \frac{1}{2} sin (2 u))))))

u = 5 x

代回

= \frac{1}{5} (\frac{sin ( 5 x ) cos ^{5} ( 5 x )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (5 x) sin (5 x) + \frac{3}{8} (5 x + \frac{1}{2} sin (2 \cdot 5 x))) - 2 (\frac{sin ( 5 x ) cos ^{7} ( 5 x )}{8} + \frac{7}{8} (\frac{sin ( 5 x ) cos ^{5} ( 5 x )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (5 x) sin (5 x) + \frac{3}{8} (5 x + \frac{1}{2} sin (2 \cdot 5 x))))) + \frac{sin ( 5 x ) cos ^{9} ( 5 x )}{10} + \frac{9}{10} (\frac{sin ( 5 x ) cos ^{7} ( 5 x )}{8} + \frac{7}{8} (\frac{sin ( 5 x ) cos ^{5} ( 5 x )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (5 x) sin (5 x) + \frac{3}{8} (5 x + \frac{1}{2} sin (2 \cdot 5 x))))))

化简

\frac{1}{5} (\frac{sin ( 5 x ) cos ^{5} ( 5 x )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (5 x) sin (5 x) + \frac{3}{8} (5 x + \frac{1}{2} sin (2 \cdot 5 x))) - 2 (\frac{sin ( 5 x ) cos ^{7} ( 5 x )}{8} + \frac{7}{8} (\frac{sin ( 5 x ) cos ^{5} ( 5 x )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (5 x) sin (5 x) + \frac{3}{8} (5 x + \frac{1}{2} sin (2 \cdot 5 x))))) + \frac{sin ( 5 x ) cos ^{9} ( 5 x )}{10} + \frac{9}{10} (\frac{sin ( 5 x ) cos ^{7} ( 5 x )}{8} + \frac{7}{8} (\frac{sin ( 5 x ) cos ^{5} ( 5 x )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (5 x) sin (5 x) + \frac{3}{8} (5 x + \frac{1}{2} sin (2 \cdot 5 x)))))) : \frac{1}{12800} (150 x + 16 cos^{5} (5 x) sin (5 x) + 20 cos^{3} (5 x) sin (5 x) + 15 sin (10 x) - 352 cos^{7} (5 x) sin (5 x) + 256 cos^{9} (5 x) sin (5 x))

= \frac{1}{12800} (150 x + 16 cos^{5} (5 x) sin (5 x) + 20 cos^{3} (5 x) sin (5 x) + 15 sin (10 x) - 352 cos^{7} (5 x) sin (5 x) + 256 cos^{9} (5 x) sin (5 x))

解答补常数

= \frac{1}{12800} (150 x + 16 cos^{5} (5 x) sin (5 x) + 20 cos^{3} (5 x) sin (5 x) + 15 sin (10 x) - 352 cos^{7} (5 x) sin (5 x) + 256 cos^{9} (5 x) sin (5 x)) + C

作图

流行的例子

derivative of-5x^{-6}

\frac{d}{d x} (- 5 x^{- 6})

integral of 11tan^3(x)sec(x)

\int 11 tan^{3} (x) sec (x) d x

derivative of (sin(5x)^{ln(x)})

\frac{d}{d x} ((sin (5 x))^{l n (x)})

derivative of (81/(x^4))

\frac{d}{d x} (\frac{81}{x ^{4}})

y^'-y/x =tan(y/x)

y^{^{'}} - \frac{y}{x} = tan (\frac{y}{x})