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受欢迎的微积分 >

integral of (3x)/((1+9x^2)^4)

解答

\int \frac{3 x}{( 1 + 9 x ^{2} ) ^{4}} d x

解答

- \frac{1}{18 ( 1 + 9 x ^{2} ) ^{3}} + C

求解步骤

\int \frac{3 x}{( 1 + 9 x ^{2} ) ^{4}} d x

提出常数:

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

= 3 \cdot \int \frac{x}{( 1 + 9 x ^{2} ) ^{4}} d x

使用换元积分法

= 3 \cdot \int \frac{1}{18 u ^{4}} d u

提出常数:

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

= 3 \cdot \frac{1}{18} \cdot \int \frac{1}{u ^{4}} d u

使用幂法则

= 3 \cdot \frac{1}{18} (- \frac{1}{3 u ^{3}})

u = 1 + 9 x^{2}

代回

= 3 \cdot \frac{1}{18} (- \frac{1}{3 ( 1 + 9 x ^{2} ) ^{3}})

化简

3 \cdot \frac{1}{18} (- \frac{1}{3 ( 1 + 9 x ^{2} ) ^{3}}) : - \frac{1}{18 ( 1 + 9 x ^{2} ) ^{3}}

= - \frac{1}{18 ( 1 + 9 x ^{2} ) ^{3}}

解答补常数

= - \frac{1}{18 ( 1 + 9 x ^{2} ) ^{3}} + C

作图

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