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受欢迎的微积分 >

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

解答

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

解答

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

求解步骤

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

对

x^{2} + x + 1

配方:

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

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

使用换元积分法

= 3 \cdot \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

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

使用换元积分法

= 3 \cdot 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

= 3 \cdot 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)

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

代回

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

化简

3 \cdot 4 \cdot \frac{1}{2 3} arctan (\frac{2}{3} (x + \frac{1}{2})) : 23 arctan (\frac{1}{3} (2 x + 1))

= 23 arctan (\frac{1}{3} (2 x + 1))

解答补常数

= 23 arctan (\frac{1}{3} (2 x + 1)) + C

作图

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