integral from 0 to pi/4 of xsec^2(x)

解答

\int_{0}^{\frac{π}{4}} x sec^{2} (x) d x

\frac{π}{4} - \frac{1}{2} ln (2)

十进制

0.43882 \dots

求解步骤

\int_{0}^{\frac{π}{4}} x sec^{2} (x) d x

使用分布积分法

= [x tan (x) - \int tan (x) d x]_{0}^{\frac{π}{4}}

\int tan (x) d x = - ln ∣ cos (x) ∣

= [x tan (x) - (- ln ∣ cos (x) ∣)]_{0}^{\frac{π}{4}}

化简

= [x tan (x) + ln ∣ cos (x) ∣]_{0}^{\frac{π}{4}}

带入上下限计算后得:

\frac{π}{4} - \frac{1}{2} ln (2)

= \frac{π}{4} - \frac{1}{2} ln (2)

d er i v a t i v e f (x) = \frac{- 4}{x}

n = 1 \sum \infty (\frac{2 π}{3})^{n}

\int \frac{1}{x ( x ^{2} + 2 )} d x

\int x cos^{2} (5 x) d x

x \to - 3 + lim (\frac{x ^{3}}{x ^{2} - 9})