题目
Problem
Find the exact value of ∫ 0 1 4 π x sec 2 x d x \displaystyle\int_0^{\frac14\pi}x\sec^2x\,\mathrm{d}x ∫ 0 4 1 π x sec 2 x d x
(5)
题目中文翻译
求 ∫ 0 1 4 π x sec 2 x d x \displaystyle\int_0^{\frac14\pi}x\sec^2x\,\mathrm{d}x ∫ 0 4 1 π x sec 2 x d x
(5)
原卷通用作答要求
Answer all questions. You should use a calculator where appropriate.
You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.
请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。
解答
解法一
思路
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分部积分时把 sec 2 x \sec^2x sec 2 x tan x \tan x tan x
答题过程
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Integrating by parts with u = x u=x u = x d v = sec 2 x d x \mathrm{d}v=\sec^2x\,\mathrm{d}x d v = sec 2 x d x
∫ x sec 2 x d x = x tan x − ∫ tan x d x . \int x\sec^2x\,\mathrm{d}x
=x\tan x-\int\tan x\,\mathrm{d}x. ∫ x sec 2 x d x = x tan x − ∫ tan x d x . Since
∫ tan x d x = ∫ sin x cos x d x = − ln ( cos x ) + C \begin{align*}
\int\tan x\,\mathrm{d}x
=&\,\int\frac{\sin x}{\cos x}\,\mathrm{d}x\\[4mm]
=&\,-\ln(\cos x)+C
\end{align*} ∫ tan x d x = = ∫ cos x sin x d x − ln ( cos x ) + C on 0 ≤ x ≤ π / 4 0\le x\le\pi/4 0 ≤ x ≤ π /4 x tan x + ln ( cos x ) x\tan x+\ln(\cos x) x tan x + ln ( cos x )
I = [ x tan x + ln ( cos x ) ] 0 π / 4 = π 4 + ln 1 2 = π 4 − 1 2 ln 2 . \begin{align*}
I
=&\,\big[x\tan x+\ln(\cos x)\big]_0^{\pi/4}\\[4mm]
=&\,\frac\pi4+\ln\frac1{\sqrt2}\\[4mm]
=&\,\boxed{\frac\pi4-\frac12\ln2}.
\end{align*} I = = = [ x tan x + ln ( cos x ) ] 0 π /4 4 π + ln 2 1 4 π − 2 1 ln 2 . 