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受欢迎的微积分 >

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

解答

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

解答

\frac{π}{3 3}

+1

十进制

0.60459 \dots

求解步骤

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

对

x^{2} + x + 1

配方:

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

= \int_{0}^{1} \frac{1}{( x + \frac{1}{2} ) ^{2} + \frac{3}{4}} d x

使用换元积分法

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

使用换元积分法

= 4 \cdot \int_{\frac{1}{3}}^{3} \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}{3}}^{3} \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)]_{\frac{1}{3}}^{3}

化简

4 \cdot \frac{1}{2 3} [arctan (v)]_{\frac{1}{3}}^{3} : \frac{2}{3} [arctan (v)]_{\frac{1}{3}}^{3}

= \frac{2}{3} [arctan (v)]_{\frac{1}{3}}^{3}

带入上下限计算后得:

\frac{π}{6}

= \frac{2}{3} \cdot \frac{π}{6}

化简

= \frac{π}{3 3}

作图

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