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受欢迎的微积分 >

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

解答

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

解答

2 (\frac{2}{3} sin^{\frac{3}{2}} (x) - \frac{2}{7} sin^{\frac{7}{2}} (x)) + C

求解步骤

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

提出常数:

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

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

使用三角恒等式改写

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

使用换元积分法

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

乘开

u (1 - u^{2}) : u - u^{\frac{5}{2}}

= 2 \cdot \int u - u^{\frac{5}{2}} 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

= 2 (\int u d u - \int u^{\frac{5}{2}} d u)

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

\int u^{\frac{5}{2}} d u = \frac{2}{7} u^{\frac{7}{2}}

= 2 (\frac{2}{3} u^{\frac{3}{2}} - \frac{2}{7} u^{\frac{7}{2}})

u = sin (x)

代回

= 2 (\frac{2}{3} sin^{\frac{3}{2}} (x) - \frac{2}{7} sin^{\frac{7}{2}} (x))

解答补常数

= 2 (\frac{2}{3} sin^{\frac{3}{2}} (x) - \frac{2}{7} sin^{\frac{7}{2}} (x)) + C

作图

流行的例子

integral of (2x)/(sqrt(x^4+9))

\int \frac{2 x}{x ^{4} + 9} d x

integral of (sin(4e^{-x}))/(e^x)

\int \frac{sin ( 4 e ^{- x} )}{e ^{x}} d x

integral of 3/5 (v-2)e^{2-4v+v^2}

\int \frac{3}{5} (v - 2) e^{2 - 4 v + v^{2}} d v

integral of sqrt(cos(θ))sin(2θ)

\int cos (θ) sin (2 θ) d θ

integral of (x^3+2x+14)(sin(3x))

\int (x^{3} + 2 x + 14) (sin (3 x)) d x