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受欢迎的微积分 >

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

解答

\int \frac{4 x}{4 x ^{2} + 1} d x

解答

\frac{1}{2} ln 4 x^{2} + 1 + C

求解步骤

\int \frac{4 x}{4 x ^{2} + 1} d x

提出常数:

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

= 4 \cdot \int \frac{x}{4 x ^{2} + 1} d x

使用换元积分法

= 4 \cdot \int \frac{1}{8 u} d u

提出常数:

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

= 4 \cdot \frac{1}{8} \cdot \int \frac{1}{u} d u

使用积分常见定则:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 4 \cdot \frac{1}{8} ln ∣ u ∣

u = 4 x^{2} + 1

代回

= 4 \cdot \frac{1}{8} ln 4 x^{2} + 1

化简

4 \cdot \frac{1}{8} ln 4 x^{2} + 1 : \frac{1}{2} ln 4 x^{2} + 1

= \frac{1}{2} ln 4 x^{2} + 1

解答补常数

= \frac{1}{2} ln 4 x^{2} + 1 + C

作图

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