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

解

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

\frac{π}{4} - \frac{1}{2} ln (2)

十進法表記

0.43882 \dots

解答ステップ

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

部分積分を適用する

= [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}}

簡素化

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

限度を計算する:

\frac{π}{4} - \frac{1}{2} ln (2)

= \frac{π}{4} - \frac{1}{2} ln (2)