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受欢迎的微积分 >

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

解答

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

解答

3 (arctan (3) - arctan (2))

+1

十进制

0.42569 \dots

求解步骤

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

提出常数:

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

= 3 \cdot \int_{1}^{2} \frac{1}{x ^{2} + 2 x + 2} d x

对

x^{2} + 2 x + 2

配方:

(x + 1)^{2} + 1

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

使用换元积分法

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

使用积分常见定则:

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

= 3 [arctan (u)]_{2}^{3}

带入上下限计算后得:

arctan (3) - arctan (2)

= 3 (arctan (3) - arctan (2))

作图

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