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受欢迎的微积分 >

integral of (x^3)/((49-4x^2)^{3/2)}

解答

\int \frac{x ^{3}}{( 49 - 4 x ^{2} ) ^{\frac{3}{2}}} d x

解答

\frac{- 2 x ^{2} + 49}{8 ( - 4 x ^{2} + 49 ) ^{\frac{1}{2}}} + C

求解步骤

\int \frac{x ^{3}}{( 49 - 4 x ^{2} ) ^{\frac{3}{2}}} d x

使用换元积分法

= \int \frac{u - 49}{32 u ^{\frac{3}{2}}} d u

提出常数:

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

= \frac{1}{32} \cdot \int \frac{u - 49}{u ^{\frac{3}{2}}} d u

乘开

\frac{u - 49}{u ^{\frac{3}{2}}} : \frac{1}{u ^{\frac{1}{2}}} - \frac{49}{u ^{\frac{3}{2}}}

= \frac{1}{32} \cdot \int \frac{1}{u ^{\frac{1}{2}}} - \frac{49}{u ^{\frac{3}{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

= \frac{1}{32} (\int \frac{1}{u ^{\frac{1}{2}}} d u - \int \frac{49}{u ^{\frac{3}{2}}} d u)

\int \frac{1}{u ^{\frac{1}{2}}} d u = 2 u^{\frac{1}{2}}

\int \frac{49}{u ^{\frac{3}{2}}} d u = - \frac{98}{u ^{\frac{1}{2}}}

= \frac{1}{32} (2 u^{\frac{1}{2}} - (- \frac{98}{u ^{\frac{1}{2}}}))

u = 49 - 4 x^{2}

代回

= \frac{1}{32} (2 (49 - 4 x^{2})^{\frac{1}{2}} - (- \frac{98}{( 49 - 4 x ^{2} ) ^{\frac{1}{2}}}))

化简

\frac{1}{32} (2 (49 - 4 x^{2})^{\frac{1}{2}} - (- \frac{98}{( 49 - 4 x ^{2} ) ^{\frac{1}{2}}})) : \frac{- 2 x ^{2} + 49}{8 ( - 4 x ^{2} + 49 ) ^{\frac{1}{2}}}

= \frac{- 2 x ^{2} + 49}{8 ( - 4 x ^{2} + 49 ) ^{\frac{1}{2}}}

解答补常数

= \frac{- 2 x ^{2} + 49}{8 ( - 4 x ^{2} + 49 ) ^{\frac{1}{2}}} + C

作图

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