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受欢迎的微积分 >

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

解答

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

解答

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

求解步骤

\int \frac{3 x}{1 + x ^{2}} 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 + x ^{2}} 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 = 1 + x^{2}

代回

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

化简

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

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

解答补常数

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

作图

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\int (- 2 x) d x

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ye^{xy}dx+xe^{xy}dy=0

y e^{x y} d x + x e^{x y} d y = 0

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