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受欢迎的微积分 >

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

解答

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

解答

ln 1 + x^{2} + C

求解步骤

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

使用换元积分法

= 2 \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

= 2 \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 ∣)

= 2 \cdot \frac{1}{2} ln ∣ u ∣

u = 1 + x^{2}

代回

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

化简

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

= ln 1 + x^{2}

解答补常数

= ln 1 + x^{2} + C

作图

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