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受欢迎的微积分 >

integral of sqrt(x^3)sin^2(1+x^{5/2})

解答

\int x^{3} sin^{2} (1 + x^{\frac{5}{2}}) d x

解答

\frac{1}{10} (2 + 2 x^{\frac{5}{2}} - sin (2 (1 + x^{\frac{5}{2}}))) + C

求解步骤

\int x^{3} sin^{2} (1 + x^{\frac{5}{2}}) d x

使用换元积分法

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

提出常数:

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

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

使用三角恒等式改写

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

提出常数:

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

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

\int 1 d u = u

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

= \frac{2}{5} \cdot \frac{1}{2} (u - \frac{1}{2} sin (2 u))

u = 1 + x^{\frac{5}{2}}

代回

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

化简

\frac{2}{5} \cdot \frac{1}{2} (1 + x^{\frac{5}{2}} - \frac{1}{2} sin (2 (1 + x^{\frac{5}{2}}))) : \frac{1}{10} (2 + 2 x^{\frac{5}{2}} - sin (2 (1 + x^{\frac{5}{2}})))

= \frac{1}{10} (2 + 2 x^{\frac{5}{2}} - sin (2 (1 + x^{\frac{5}{2}})))

解答补常数

= \frac{1}{10} (2 + 2 x^{\frac{5}{2}} - sin (2 (1 + x^{\frac{5}{2}}))) + C

作图

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