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受欢迎的微积分 >

integral of (4x)/(2x^2+5)

解答

\int \frac{4 x}{2 x ^{2} + 5} d x

解答

ln 2 x^{2} + 5 + C

求解步骤

\int \frac{4 x}{2 x ^{2} + 5} 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}{2 x ^{2} + 5} d x

使用换元积分法

= 4 \cdot \int \frac{1}{4 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}{4} \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}{4} ln ∣ u ∣

u = 2 x^{2} + 5

代回

= 4 \cdot \frac{1}{4} ln 2 x^{2} + 5

化简

4 \cdot \frac{1}{4} ln 2 x^{2} + 5 : ln 2 x^{2} + 5

= ln 2 x^{2} + 5

解答补常数

= ln 2 x^{2} + 5 + C

作图

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