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受欢迎的微积分 >

integral of 6x^3sqrt(9-x^2)

解答

\int 6 x^{3} 9 - x^{2} d x

解答

- 6 (- \frac{1}{5} (9 - x^{2})^{\frac{5}{2}} + 3 (9 - x^{2})^{\frac{3}{2}}) + C

求解步骤

\int 6 x^{3} 9 - x^{2} d x

提出常数:

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

= 6 \cdot \int x^{3} 9 - x^{2} d x

使用换元积分法

= 6 \cdot \int - (- u^{2} + 9) u^{2} d u

提出常数:

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

= 6 (- \int (- u^{2} + 9) u^{2} d u)

乘开

(- u^{2} + 9) u^{2} : - u^{4} + 9 u^{2}

= 6 (- \int - u^{4} + 9 u^{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

= 6 (- (- \int u^{4} d u + \int 9 u^{2} d u))

\int u^{4} d u = \frac{u ^{5}}{5}

\int 9 u^{2} d u = 3 u^{3}

= 6 (- (- \frac{u ^{5}}{5} + 3 u^{3}))

u = 9 - x^{2}

代回

= 6 (- (- \frac{( 9 - x ^{2} ) ^{5}}{5} + 3 (9 - x^{2})^{3}))

化简

6 (- (- \frac{( 9 - x ^{2} ) ^{5}}{5} + 3 (9 - x^{2})^{3})) : - 6 (- \frac{1}{5} (9 - x^{2})^{\frac{5}{2}} + 3 (9 - x^{2})^{\frac{3}{2}})

= - 6 (- \frac{1}{5} (9 - x^{2})^{\frac{5}{2}} + 3 (9 - x^{2})^{\frac{3}{2}})

解答补常数

= - 6 (- \frac{1}{5} (9 - x^{2})^{\frac{5}{2}} + 3 (9 - x^{2})^{\frac{3}{2}}) + C

作图

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