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受欢迎的微积分 >

integral of 1/(2x^2+8x+20)

解答

\int \frac{1}{2 x ^{2} + 8 x + 20} d x

解答

\frac{1}{2 6} arctan (\frac{1}{6} (x + 2)) + C

求解步骤

\int \frac{1}{2 x ^{2} + 8 x + 20} d x

对

2 x^{2} + 8 x + 20

配方:

2 (x + 2)^{2} + 12

= \int \frac{1}{2 ( x + 2 ) ^{2} + 12} d x

使用换元积分法

= \int \frac{1}{2 u ^{2} + 12} d u

使用换元积分法

= \int \frac{1}{2 6 ( v ^{2} + 1 )} d v

提出常数:

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

= \frac{1}{2 6} \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)

= \frac{1}{2 6} arctan (v)

代回

= \frac{1}{2 6} arctan (\frac{2}{2 3} (x + 2))

化简

\frac{1}{2 6} arctan (\frac{2}{2 3} (x + 2)) : \frac{1}{2 6} arctan (\frac{1}{6} (x + 2))

= \frac{1}{2 6} arctan (\frac{1}{6} (x + 2))

解答补常数

= \frac{1}{2 6} arctan (\frac{1}{6} (x + 2)) + C

作图

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