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受欢迎的微积分 >

integral of 1/(x^2+6x+12)

解答

\int \frac{1}{x ^{2} + 6 x + 12} d x

解答

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

求解步骤

\int \frac{1}{x ^{2} + 6 x + 12} d x

对

x^{2} + 6 x + 12

配方:

(x + 3)^{2} + 3

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

使用换元积分法

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

使用换元积分法

= \int \frac{1}{3 ( 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}{3} \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}{3} arctan (v)

代回

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

解答补常数

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

作图

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