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受欢迎的微积分 >

integral of (-3x)/(x^2+1)

解答

\int \frac{- 3 x}{x ^{2} + 1} d x

解答

- \frac{3}{2} ln x^{2} + 1 + C

求解步骤

\int \frac{- 3 x}{x ^{2} + 1} 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}{x ^{2} + 1} d x

使用换元积分法

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

提出常数:

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

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

使用积分常见定则:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - 3 \cdot \frac{1}{2} ln ∣ u ∣

u = x^{2} + 1

代回

= - 3 \cdot \frac{1}{2} ln x^{2} + 1

化简

- 3 \cdot \frac{1}{2} ln x^{2} + 1 : - \frac{3}{2} ln x^{2} + 1

= - \frac{3}{2} ln x^{2} + 1

解答补常数

= - \frac{3}{2} ln x^{2} + 1 + C

作图

流行的例子

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\int \frac{5}{x ^{3}} + \frac{5}{x ^{2}} + \frac{5}{x} d x

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

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\int \frac{5 x}{x ^{2} + 3} d x

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\int \frac{x}{16 - 4 x ^{2}} d x

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\int \frac{1}{4 x ^{7}} d x