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受欢迎的微积分 >

integral of sqrt(x)sqrt(9-x)

解答

\int x 9 - x d x

解答

\frac{81}{16} (2 arcsin (\frac{2}{9} x - 1) + sin (2 arcsin (\frac{2}{9} x - 1))) + C

求解步骤

\int x 9 - x d x

使用指数法则:

a^{n} \cdot b^{n} = (a \cdot b)^{n}

x 9 - x = x (9 - x)

= \int x (9 - x) d x

乘开

x (9 - x) : 9 x - x^{2}

= \int 9 x - x^{2} d x

对

9 x - x^{2}

配方:

- (x - \frac{9}{2})^{2} + \frac{81}{4}

= \int - (x - \frac{9}{2})^{2} + \frac{81}{4} d x

使用换元积分法

= \int \frac{- 4 u ^{2} + 81}{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 - 4 u^{2} + 81 d u

使用三角换元法

= \frac{1}{2} \cdot \int \frac{81}{2} cos^{2} (v) d v

提出常数:

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

= \frac{1}{2} \cdot \frac{81}{2} \cdot \int cos^{2} (v) d v

使用三角恒等式改写

= \frac{1}{2} \cdot \frac{81}{2} \cdot \int \frac{1 + cos ( 2 v )}{2} d v

提出常数:

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

= \frac{1}{2} \cdot \frac{81}{2} \cdot \frac{1}{2} \cdot \int 1 + cos (2 v) 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} \cdot \frac{81}{2} \cdot \frac{1}{2} (\int 1 d v + \int cos (2 v) d v)

\int 1 d v = v

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

= \frac{1}{2} \cdot \frac{81}{2} \cdot \frac{1}{2} (v + \frac{1}{2} sin (2 v))

代回

= \frac{1}{2} \cdot \frac{81}{2} \cdot \frac{1}{2} (arcsin (\frac{2}{9} (x - \frac{9}{2})) + \frac{1}{2} sin (2 arcsin (\frac{2}{9} (x - \frac{9}{2}))))

化简

\frac{1}{2} \cdot \frac{81}{2} \cdot \frac{1}{2} (arcsin (\frac{2}{9} (x - \frac{9}{2})) + \frac{1}{2} sin (2 arcsin (\frac{2}{9} (x - \frac{9}{2})))) : \frac{81}{16} (2 arcsin (\frac{2}{9} x - 1) + sin (2 arcsin (\frac{2}{9} x - 1)))

= \frac{81}{16} (2 arcsin (\frac{2}{9} x - 1) + sin (2 arcsin (\frac{2}{9} x - 1)))

解答补常数

= \frac{81}{16} (2 arcsin (\frac{2}{9} x - 1) + sin (2 arcsin (\frac{2}{9} x - 1))) + C

作图

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