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受欢迎的微积分 >

integral of (2x)/((11-5x^2)^4)

解答

\int \frac{2 x}{( 11 - 5 x ^{2} ) ^{4}} d x

解答

- \frac{1}{15 ( 5 x ^{2} - 11 ) ^{3}} + C

求解步骤

\int \frac{2 x}{( 11 - 5 x ^{2} ) ^{4}} d x

提出常数:

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

= 2 \cdot \int \frac{x}{( 11 - 5 x ^{2} ) ^{4}} d x

因式分解

11 - 5 x^{2} : - (- 11 + 5 x^{2})

= 2 \cdot \int \frac{x}{( - ( - 11 + 5 x ^{2} ) ) ^{4}} d x

(- (- 11 + 5 x^{2}))^{4} = (- 11 + 5 x^{2})^{4}

= 2 \cdot \int \frac{x}{( - 11 + 5 x ^{2} ) ^{4}} d x

使用换元积分法

= 2 \cdot \int \frac{1}{10 u ^{4}} d u

提出常数:

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

= 2 \cdot \frac{1}{10} \cdot \int \frac{1}{u ^{4}} d u

使用幂法则

= 2 \cdot \frac{1}{10} (- \frac{1}{3 u ^{3}})

u = - 11 + 5 x^{2}

代回

= 2 \cdot \frac{1}{10} (- \frac{1}{3 ( - 11 + 5 x ^{2} ) ^{3}})

化简

2 \cdot \frac{1}{10} (- \frac{1}{3 ( - 11 + 5 x ^{2} ) ^{3}}) : - \frac{1}{15 ( 5 x ^{2} - 11 ) ^{3}}

= - \frac{1}{15 ( 5 x ^{2} - 11 ) ^{3}}

解答补常数

= - \frac{1}{15 ( 5 x ^{2} - 11 ) ^{3}} + C

作图

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