integral of (5x^3)/(sqrt(49-x^2))

解答

\int \frac{5 x ^{3}}{49 - x ^{2}} d x

5 (\frac{1}{3} (49 - x^{2})^{\frac{3}{2}} - 49 49 - x^{2}) + C

求解步骤

\int \frac{5 x ^{3}}{49 - x ^{2}} d x

提出常数:

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

= 5 \cdot \int \frac{x ^{3}}{49 - x ^{2}} d x

使用换元积分法

= 5 \cdot \int u^{2} - 49 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

= 5 (\int u^{2} d u - \int 49 d u)

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

\int 49 d u = 49 u

= 5 (\frac{u ^{3}}{3} - 49 u)

u = 49 - x^{2}

代回

= 5 (\frac{( 49 - x ^{2} ) ^{3}}{3} - 49 49 - x^{2})

化简

5 (\frac{( 49 - x ^{2} ) ^{3}}{3} - 49 49 - x^{2}) : 5 (\frac{1}{3} (49 - x^{2})^{\frac{3}{2}} - 49 49 - x^{2})

= 5 (\frac{1}{3} (49 - x^{2})^{\frac{3}{2}} - 49 49 - x^{2})

解答补常数

= 5 (\frac{1}{3} (49 - x^{2})^{\frac{3}{2}} - 49 49 - x^{2}) + C

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