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受欢迎的微积分 >

integral of sin^2(3x)*cos^4(3x)

解答

\int sin^{2} (3 x) \cdot cos^{4} (3 x) d x

解答

\frac{1}{288} (18 x + 76 cos^{3} (3 x) sin (3 x) - 9 sin (12 x) - 16 cos^{5} (3 x) sin (3 x) - 15 sin (6 x)) + C

求解步骤

\int sin^{2} (3 x) cos^{4} (3 x) d x

使用换元积分法

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

提出常数:

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

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

使用三角恒等式改写

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

乘开

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

= \frac{1}{3} \cdot \int cos^{4} (u) - cos^{6} (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}{3} (\int cos^{4} (u) d u - \int cos^{6} (u) d u)

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

= \frac{1}{3} (cos^{3} (u) sin (u) + \frac{3}{8} (u - \frac{1}{4} sin (4 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)))))

u = 3 x

代回

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

化简

\frac{1}{3} (cos^{3} (3 x) sin (3 x) + \frac{3}{8} (3 x - \frac{1}{4} sin (4 \cdot 3 x)) - (\frac{sin ( 3 x ) cos ^{5} ( 3 x )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (3 x) sin (3 x) + \frac{3}{8} (3 x + \frac{1}{2} sin (2 \cdot 3 x))))) : \frac{1}{288} (18 x + 76 cos^{3} (3 x) sin (3 x) - 9 sin (12 x) - 16 cos^{5} (3 x) sin (3 x) - 15 sin (6 x))

= \frac{1}{288} (18 x + 76 cos^{3} (3 x) sin (3 x) - 9 sin (12 x) - 16 cos^{5} (3 x) sin (3 x) - 15 sin (6 x))

解答补常数

= \frac{1}{288} (18 x + 76 cos^{3} (3 x) sin (3 x) - 9 sin (12 x) - 16 cos^{5} (3 x) sin (3 x) - 15 sin (6 x)) + C

作图

流行的例子

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\int \frac{2 x ^{2} - 5 x - 18}{x ^{3} + x ^{2} - 6 x} d x

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\int \frac{r ^{2}}{r + 4} d r

integral from 0 to pi/4 of (sec^2(x))

\int_{0}^{\frac{π}{4}} (sec^{2} (x)) d x

integral of x^5cos(1-x^6)

\int x^{5} cos (1 - x^{6}) d x

limit as x approaches 0 of 5x-3

x \to 0 lim (5 x - 3)