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受欢迎的微积分 >

integral from 3 to 4 of 1/(25-x^2)

解答

\int_{3}^{4} \frac{1}{25 - x ^{2}} d x

解答

\frac{ln ( \frac{9}{4} )}{10}

+1

十进制

0.08109 \dots

求解步骤

\int_{3}^{4} \frac{1}{25 - x ^{2}} d x

使用换元积分法

= \int_{\frac{3}{5}}^{\frac{4}{5}} \frac{1}{5 ( - 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}{5} \cdot \int_{\frac{3}{5}}^{\frac{4}{5}} \frac{1}{- u ^{2} + 1} d u

使用积分常见定则:

\int \frac{1}{- u ^{2} + 1} d u = \frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}

= \frac{1}{5} [\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}]_{\frac{3}{5}}^{\frac{4}{5}}

化简

\frac{1}{5} [\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}]_{\frac{3}{5}}^{\frac{4}{5}} : \frac{1}{5} [\frac{ln ∣ u + 1 ∣ - ln ∣ u - 1 ∣}{2}]_{\frac{3}{5}}^{\frac{4}{5}}

= \frac{1}{5} [\frac{ln ∣ u + 1 ∣ - ln ∣ u - 1 ∣}{2}]_{\frac{3}{5}}^{\frac{4}{5}}

带入上下限计算后得:

\frac{ln ( \frac{9}{5} ) + ln ( 5 ) - ln ( \frac{8}{5} ) + ln ( \frac{2}{5} )}{2}

= \frac{1}{5} \cdot \frac{ln ( \frac{9}{5} ) + ln ( 5 ) - ln ( \frac{8}{5} ) + ln ( \frac{2}{5} )}{2}

化简

= \frac{ln ( \frac{9}{4} )}{10}

作图

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