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受欢迎的微积分 >

integral from 0 to pi/3 of x/(cos^2(x))

解答

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

解答

\frac{π}{3} - ln (2)

+1

十进制

1.12065 \dots

求解步骤

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

使用三角恒等式改写

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

提出常数:

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

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

使用分布积分法

= 2 [\frac{1}{2} (x tan (x) - 2 \cdot \int \frac{1}{2} tan (x) d x)]_{0}^{\frac{π}{3}}

\int \frac{1}{2} tan (x) d x = - \frac{1}{2} ln ∣ cos (x) ∣

= 2 [\frac{1}{2} (x tan (x) - 2 (- \frac{1}{2} ln ∣ cos (x) ∣))]_{0}^{\frac{π}{3}}

化简

2 [\frac{1}{2} (x tan (x) - 2 (- \frac{1}{2} ln ∣ cos (x) ∣))]_{0}^{\frac{π}{3}} : 2 [\frac{1}{2} (x tan (x) + ln ∣ cos (x) ∣)]_{0}^{\frac{π}{3}}

= 2 [\frac{1}{2} (x tan (x) + ln ∣ cos (x) ∣)]_{0}^{\frac{π}{3}}

带入上下限计算后得:

\frac{1}{2} (\frac{π}{3} - ln (2))

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

化简

= \frac{π}{3} - ln (2)

作图

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