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受欢迎的微积分 >

integral of 3/(2x^2-4x+10)

解答

\int \frac{3}{2 x ^{2} - 4 x + 10} d x

解答

\frac{3}{4} arctan (\frac{x - 1}{2}) + C

求解步骤

\int \frac{3}{2 x ^{2} - 4 x + 10} d x

提出常数:

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

= 3 \cdot \int \frac{1}{2 x ^{2} - 4 x + 10} d x

对

2 x^{2} - 4 x + 10

配方:

2 (x - 1)^{2} + 8

= 3 \cdot \int \frac{1}{2 ( x - 1 ) ^{2} + 8} d x

使用换元积分法

= 3 \cdot \int \frac{1}{2 u ^{2} + 8} d u

使用换元积分法

= 3 \cdot \int \frac{1}{4 ( v ^{2} + 1 )} d v

提出常数:

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

= 3 \cdot \frac{1}{4} \cdot \int \frac{1}{v ^{2} + 1} d v

使用积分常见定则:

\int \frac{1}{v ^{2} + 1} d v = arctan (v)

= 3 \cdot \frac{1}{4} arctan (v)

代回

= 3 \cdot \frac{1}{4} arctan (\frac{2}{2 2} (x - 1))

化简

3 \cdot \frac{1}{4} arctan (\frac{2}{2 2} (x - 1)) : \frac{3}{4} arctan (\frac{x - 1}{2})

= \frac{3}{4} arctan (\frac{x - 1}{2})

解答补常数

= \frac{3}{4} arctan (\frac{x - 1}{2}) + C

作图

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