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【笔记】高数学习笔记 - 区间再现

panedioic
2022-09-27 / 0 评论 / 0 点赞 / 193 阅读 / 1,070 字

一、区间再现的表达

abf(x)dx=abf(a+bx)dx=12ab[f(x)+f(a+bx)]dx\int_a^bf(x)dx = \int_a^bf(a+b-x)dx = \frac{1}{2}\int_a^b[f(x)+f(a+b-x)]dx

证明:

abf(a+bx)dx=t=a+bxbaf(t)(1)dt=abf(t)dt=abf(x)dx\int_a^bf(a+b-x)dx\overset{令t=a+b-x}{=}\int_b^af(t)(-1)dt=\int_a^bf(t)dt=\int_a^bf(x)dx

例题

1.10πsinxsinx+cosxdx2.10π211+(tanx)αdx2.20+inf1(1+x2)(1+xα)dx3.1π2π2(arctanex)sin2xdx4.122xln(1+ex)dx5.1π6π3cos2xx(π2x)dx6.00π4xcos(π4x)cosxdx6.11sin(x+a)sin(x+b)6.21sin(x+3)sin(x+5)7.00π2lnsinxdx7.10π2xtanxdx7.2π2π2cosxlncosx1+sinx+cosxdx8.001ln(1+x)1+x2dx8.101arctanx1+xdx9.00nπxsinxdx9.1an=0nπxsinxdx,n=1inf1an10.001arcsinxx2x+1dx11.001xarcsin2xx2dx12.001(1x)100xdx13.002x(x1)(x2)dx13.102nx(x1)(x2)(x2n)dx\begin{aligned} &1.1\qquad \int_0^\pi \frac{\sin x}{\sin x+\cos x}dx \\ &2.1\qquad \int_0^{\frac{\pi}{2}}\frac{1}{1+(\tan x)^\alpha}dx \\ &2.2\qquad \int_0^{+\inf}\frac{1}{(1+x^2)(1+x^{\alpha})}dx \\ &3.1\qquad \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}(\arctan e^x)\cdot \sin^2xdx \\ &4.1\qquad \int_{-2}^{2}x\cdot\ln(1+e^x)dx \\ &5.1\qquad \int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\frac{\cos^2x}{x(\pi-2x)}dx \\ &6.0\qquad \int_0^{\frac{\pi}{4}}\frac{x}{\cos(\frac{\pi}{4}-x)\cdot\cos x}dx \\ &6.1\qquad \int\frac{1}{\sin(x+a)\cdot\sin(x+b)} \\ &6.2\qquad \int\frac{1}{\sin(x+3)\cdot\sin(x+5)} \\ &7.0\qquad \int_0^{\frac{\pi}{2}}\ln\sin xdx \\ &7.1\qquad\int_0^{\frac{\pi}{2}}\frac{x}{\tan x}dx \\ &7.2\qquad \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{\cos x\ln \cos x}{1+\sin x+\cos x}dx \\ &8.0\qquad \int_0^1\frac{\ln(1+x)}{1+x^2}dx \\ &8.1\qquad \int_0^1\frac{\arctan x}{1+x}dx \\ &9.0\qquad \int_0^{n\pi}x\cdot |\sin x| dx \\ &9.1\qquad a_n=\int_0^{n\pi}x\cdot |\sin x| dx, 求\sum_{n=1}^{\inf}\frac{1}{a_n} \\ &10.0\qquad \int_0^1\frac{\arcsin\sqrt{x}}{\sqrt{x^2-x+1}}dx \\ &11.0\qquad \int_0^1x\cdot\arcsin 2\sqrt{x-x^2}dx \\ &12.0\qquad \int_0^1(1-x)^{100}\cdot xdx \\ &13.0\qquad \int_0^2x\cdot(x-1)\cdot(x-2)dx \\ &13.1\qquad \int_0^{2n}x\cdot(x-1)\cdot(x-2)\cdot\cdot\cdot\cdot(x-2n) dx \end{aligned} \\

例题解析

之后再补吧。
这是看凯哥的学习笔记。原视频可以参考:
https://www.bilibili.com/video/BV1ah411Y789

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