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受欢迎的微积分 >

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

解答

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

解答

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

求解步骤

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

对

x^{2} + 3 x + 3

配方:

(x + \frac{3}{2})^{2} + \frac{3}{4}

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

使用换元积分法

= \int \frac{4}{4 u ^{2} + 3} d u

提出常数:

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

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

使用换元积分法

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

提出常数:

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

= 4 \cdot \frac{1}{2 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)

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

代回

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

化简

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

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

解答补常数

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

作图

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