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受欢迎的微积分 >

integral from-2 to 0 of (x+2)/(x^2+2x+2)

解答

\int_{- 2}^{0} \frac{x + 2}{x ^{2} + 2 x + 2} d x

解答

\frac{π}{2}

+1

十进制

1.57079 \dots

求解步骤

\int_{- 2}^{0} \frac{x + 2}{x ^{2} + 2 x + 2} d x

对

x^{2} + 2 x + 2

配方:

(x + 1)^{2} + 1

= \int_{- 2}^{0} \frac{x + 2}{( x + 1 ) ^{2} + 1} d x

使用换元积分法

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

乘开

\frac{u + 1}{u ^{2} + 1} : \frac{u}{u ^{2} + 1} + \frac{1}{u ^{2} + 1}

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int_{- 1}^{1} \frac{u}{u ^{2} + 1} d u = 0

\int_{- 1}^{1} \frac{1}{u ^{2} + 1} d u = \frac{π}{2}

= 0 + \frac{π}{2}

化简

= \frac{π}{2}

作图

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