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受欢迎的微积分 >

integral from 0 to pi/2 of 1/(x^2+2)

解答

\int_{0}^{\frac{π}{2}} \frac{1}{x ^{2} + 2} d x

解答

\frac{arctan ( \frac{π}{2 2} )}{2}

+1

十进制

0.59241 \dots

求解步骤

\int_{0}^{\frac{π}{2}} \frac{1}{x ^{2} + 2} d x

使用换元积分法

= \int_{0}^{\frac{π}{2 2}} \frac{1}{2 ( u ^{2} + 1 )} d u

提出常数:

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

= \frac{1}{2} \cdot \int_{0}^{\frac{π}{2 2}} \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)

= \frac{1}{2} [arctan (u)]_{0}^{\frac{π}{2 2}}

带入上下限计算后得:

arctan (\frac{π}{2 2})

= \frac{1}{2} arctan (\frac{π}{2 2})

化简

= \frac{arctan ( \frac{π}{2 2} )}{2}

作图

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