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受欢迎的微积分 >

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

解答

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

解答

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

求解步骤

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

对

x^{2} + 2 x + 3

配方:

(x + 1)^{2} + 2

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

使用换元积分法

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

使用换元积分法

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

代回

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

解答补常数

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

作图

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