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受欢迎的微积分 >

integral from 0 to pi/2 of 1/(3+5cos(x))

解答

\int_{0}^{\frac{π}{2}} \frac{1}{3 + 5 cos ( x )} d x

解答

\frac{ln ( 3 )}{4}

+1

十进制

0.27465 \dots

求解步骤

\int_{0}^{\frac{π}{2}} \frac{1}{3 + 5 cos ( x )} d x

使用换元积分法

= \int_{0}^{1} \frac{1}{- u ^{2} + 4} d u

使用换元积分法

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

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{1}{2}} \frac{1}{- v ^{2} + 1} d v

使用积分常见定则:

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

= \frac{1}{2} [\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}]_{0}^{\frac{1}{2}}

化简

\frac{1}{2} [\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}]_{0}^{\frac{1}{2}} : \frac{1}{2} [\frac{ln ∣ v + 1 ∣ - ln ∣ v - 1 ∣}{2}]_{0}^{\frac{1}{2}}

= \frac{1}{2} [\frac{ln ∣ v + 1 ∣ - ln ∣ v - 1 ∣}{2}]_{0}^{\frac{1}{2}}

带入上下限计算后得:

\frac{ln ( 2 ) + ln ( \frac{3}{2} )}{2}

= \frac{1}{2} \cdot \frac{ln ( 2 ) + ln ( \frac{3}{2} )}{2}

化简

= \frac{ln ( 3 )}{4}

作图

流行的例子

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