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受欢迎的微积分 >

integral of x/((sqrt(7^2-x^2))^3)

解答

\int \frac{x}{( 7 ^{2} - x ^{2} ) ^{3}} d x

解答

\frac{1}{49 - x ^{2}} + C

求解步骤

\int \frac{x}{( 7 ^{2} - x ^{2} ) ^{3}} d x

化简

\frac{x}{( 7 ^{2} - x ^{2} ) ^{3}} : \frac{x}{( 49 - x ^{2} ) ^{\frac{3}{2}}}

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

使用换元积分法

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

使用幂法则

= - \frac{1}{2} (- \frac{2}{u})

u = 49 - x^{2}

代回

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

化简

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

= \frac{1}{49 - x ^{2}}

解答补常数

= \frac{1}{49 - x ^{2}} + C

作图

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