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受欢迎的微积分 >

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

解答

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

解答

\frac{1}{2} (\frac{sin ^{3} ( 2 x )}{3} - \frac{sin ^{5} ( 2 x )}{5}) + C

求解步骤

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

使用换元积分法

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

提出常数:

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

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

使用三角恒等式改写

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

使用换元积分法

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

乘开

v^{2} (1 - v^{2}) : v^{2} - v^{4}

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

\int v^{2} d v = \frac{v ^{3}}{3}

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

= \frac{1}{2} (\frac{v ^{3}}{3} - \frac{v ^{5}}{5})

代回

= \frac{1}{2} (\frac{sin ^{3} ( 2 x )}{3} - \frac{sin ^{5} ( 2 x )}{5})

解答补常数

= \frac{1}{2} (\frac{sin ^{3} ( 2 x )}{3} - \frac{sin ^{5} ( 2 x )}{5}) + C

作图

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\int ln^{4} (x) d x

d/(dt)(sin(pi)t)

\frac{d}{d t} (sin (π) t)

d/(dy)(A/(y^7)+Be^y)

\frac{d}{d y} (\frac{A}{y ^{7}} + B e^{y})

integral of 6sqrt(x)

\int 6 x d x

derivative f(x)=(x^3+2x^2-4)/3

d er i v a t i v e f (x) = \frac{x ^{3} + 2 x ^{2} - 4}{3}