Ch-6 定积分

定积分中值定理

若 \(f(x)\) 在 \([a,b]\) 上可积，则 \(\exists\xi\in(a,b)\) ，使得 \(\begin{aligned}\int_a^bf(x)\mathrm{d}x=f(\xi)(b-a)\end{aligned}\)

推广的定积分中值定理

若 \(f(x),g(x)\) 在 \([a,b]\) 上连续，且 \(g(x)\) 在 \([a,b]\) 上不变号，则 \(\exists\xi\in(a,b)\)，使得

\(\begin{aligned}\int_a^bf(x)g(x)\mathrm{d}x=f(\xi)\int_a^bg(x)\mathrm{d}x\end{aligned}\)

柯西不等式

\(\begin{pmatrix}\begin{aligned}\int_a^bf(x)g(x)\mathrm{d}x\end{aligned}\end{pmatrix}^2\leq\begin{aligned}\int_a^bf^2(x)\mathrm{d}x\cdot\int_a^bg^2(x)\mathrm{d}x\end{aligned}\)

重要结果及技巧

\(\begin{aligned}\int\csc{x}\mathrm{d}x=\int\frac{\csc{x}(\csc{x}-\cot{x})}{\csc{x}-\cot{x}}\mathrm{d}x=\int\frac{\mathrm{d}(\csc{x}-\cot{x})}{\csc{x}-\cot{x}}=\ln|\csc{x}-\cot{x}|\end{aligned}\)

\(\begin{aligned}\int\sec{x}\mathrm{d}x=\int\frac{\sec{x}(\sec{x}+\tan{x})}{\sec{x}+\tan{x}}\mathrm{d}x=\int\frac{\mathrm{d}(\sec x+\tan x)}{\sec x+\tan x}=\ln|\sec x+\tan x|\end{aligned}\)

\(\begin{aligned}\int\cos^nx\mathrm{d}x=\frac{1}{n}\sin x\cos^{n-1}x+\frac{n-1}{n}\int\cos^{n-2}x\mathrm{d}x\end{aligned}\)

\(\begin{aligned}\int\sin^nx\mathrm{d}x=-\frac{1}{n}\cos x\sin^{n-1}x+\frac{n-1}{n}\int\sin^{n-2}x\mathrm{d}x\end{aligned}\)

\(\begin{aligned}I_n=\int\frac{\mathrm{d}x}{(a^2+x^2)^n},I_n=\frac{x}{2(n-1)a^2(a^2+x^2)^{n-1}}+\frac{2n-3}{2(n-1)a^2}I_{n-1}\end{aligned}\)

\(\mathrm{d}(\sin x+\cos x)=\cos x-\sin x,\mathrm{d}(\sin x-\cos x)=\sin x+\cos x\)

\(\begin{aligned}\int_0^{\frac{\pi}{2}}\sin^nx\mathrm{d}x=\int_0^{\frac{\pi}{2}}\cos^nx\mathrm{d}x\end{aligned}=\begin{cases}\dfrac{n-1}{n}\cdot\dfrac{n-3}{n-2}\cdot\cdots\cdot\dfrac{2}{3}\cdot1,&n为奇数\\\dfrac{n-1}{n}\cdot\dfrac{n-3}{n-2}\cdot\cdots\cdot\dfrac{3}{4}\cdot\dfrac{1}{2}\cdot\dfrac{\pi}{2},&n为偶数\end{cases}\)

\(\begin{aligned}\Gamma(s+1)&=\int_0^{+\infty}x^{s}e^{-x}\mathrm{d}x=-x^se^{-x}\big|_0^{+\infty}+s\int_0^{+\infty}x^{s-1}e^{-x}\mathrm{d}x\\&=s\Gamma(s)=\cdots=n!\Gamma(1)=n!\end{aligned}\)

对于一切 \(a>0\) ，极限 \(\begin{aligned}\lim_{x\rightarrow0^+}x^a\ln x=0\end{aligned}\) 成立

\(\begin{aligned}\lim_{x\rightarrow0^+}x^a\ln x=\lim_{x\rightarrow0^+}\dfrac{\ln x}{x^{-a}}=-\frac{1}{a}\lim_{x\rightarrow0^+}\frac{x^{-1}}{x^{-a-1}}=-\frac{1}{a}\lim_{x\rightarrow0^+}x^a=0\end{aligned}\)

在区间对称的积分中分离出奇函数的部分

中值定理构造原函数

将结果中的未知参数换为 \(x\)

如果结果中既有 \(f(x)\) 又含有 \(f'(x)\) ，则将等式变换为 \(\dfrac{f'(x)}{f(x)}+h(x)=0\) 的形式

假使 \(p'(x)=h(x)\) ，则构造 \(g(x)=f(x)e^{p(x)}\)

积分上限函数

\(\begin{bmatrix}\begin{aligned}\int_a^{g(x)}f(t)\mathrm{d}t\end{aligned}\end{bmatrix}'=f[g(x)]g'(x)\)

\(\begin{bmatrix}\begin{aligned}\int_{h(x)}^{g(x)}f(t)\mathrm{d}t\end{aligned}\end{bmatrix}'=f[g(x)]g'(x)-f[h(x)]h'(x)\)