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z=vb!L$0DcJ4=-F@9^>1<*Vk2oAe%v(w#nB@)3JAXublu?yp&F7?dM1%-BzDK9u+kK zdr85rB%RXWXxWmd#Sz>25wPO9WXH zjf2DqYZ_?fwhy6Rj7o*XL&mh)ad6VtB- zswZ=|TA3o_Nhi;B{|<}-Z@af>_y7?N3wb-opmCKG86VqQ$orzgC2#ePV{rDIZY6i9 v$vh;jD-_)xtD_;rGv&*4sJKti=i&wK6e44Ae0U~yE;cS$GBOc4QP}?nmPJ!t diff --git a/advanced-math/exercise/4-integal-of-functions-of-single-variable/integal-of-functions-of-single-variable.tex b/advanced-math/exercise/4-integal-of-functions-of-single-variable/integal-of-functions-of-single-variable.tex index bd0bde0..9750b1c 100644 --- a/advanced-math/exercise/4-integal-of-functions-of-single-variable/integal-of-functions-of-single-variable.tex +++ b/advanced-math/exercise/4-integal-of-functions-of-single-variable/integal-of-functions-of-single-variable.tex @@ -709,6 +709,19 @@ $=\int_{-\frac{\pi}{4}}^\frac{\pi}{4}\cos^2u\,\textrm{d}u+\int_{-\frac{\pi}{4}}^ \textbf{例题:}求$\int_0^\frac{\pi}{2}e^{2x}\cos x\,\textrm{d}x$。 +解:令$e^{2x}=u$,$\cos x=v$,$\therefore$根据积分推广公式:\medskip + +\begin{tabular}{|c|c|c|} + \hline + $e^{2x}$ & $2e^{2x}$ & $4e^{2x}$ \\ \hline + $\cos x$ & $\sin x$ & $-\cos x$ \\ + \hline +\end{tabular} \medskip + +$\therefore\int_0^\frac{\pi}{2}e^{2x}\cos x\,\textrm{d}x=[e^{2x}\sin x+2e^{2x}\cos x]_0^\frac{\pi}{2}-4\int_0^\frac{\pi}{2}e^{2x}\cos x\,\textrm{d}x$。 + +$\int_0^\frac{\pi}{2}e^{2x}\cos x\,\textrm{d}x=\dfrac{1}{5}[e^{2x}(\sin x+2\cos x)]_0^\frac{\pi}{2}=\dfrac{1}{5}(e^\pi-2)$。 + \subsection{变限积分} \subsubsection{极限} @@ -747,6 +760,32 @@ $\therefore\varPhi(x)=\left\{\begin{array}{ll} \subsection{反常积分} +反常积分就是取极限,基本计算方法一样。 + +\subsubsection{基本反常积分} + +\textbf{例题:}求$\displaystyle{\int_0^{+\infty}\dfrac{\textrm{d}x}{(1+x)(1+x^2)}}$。\medskip + +解:这是无穷区间的反常积分,看到一个$1+x^2$因式,所以尝试使用换元积分法,令$x=\tan u$,$1+x^2=\sec^2u$,$\textrm{d}x=\sec^2u\,\textrm{d}u$。 + +当$x=0$,$\tan u=0$,$u=0$,当$x=+\infty$,$u=+\dfrac{\pi}{2}$。 + +$\therefore=\displaystyle{\int_0^\frac{\pi}{2}\dfrac{1}{1+\tan u}\textrm{d}u=\int_0^\frac{\pi}{2}\dfrac{\cos u}{\cos u+\sin u}\textrm{d}u=\dfrac{\pi}{4}}$。 + +根据区间再现公式,$\displaystyle{\int_0^\frac{\pi}{2}\dfrac{\sin u}{\cos u+\sin u}\textrm{d}u=\int_0^\frac{\pi}{2}\dfrac{\cos u}{\cos u+\sin u}\textrm{d}u=\dfrac{\pi}{4}}$。 + +\subsubsection{递推公式} + +\textbf{例题:}利用递推公式计算反常积分$I_n=\int_0^{+\infty}x^ne^{-x}\,\textrm{d}x$($n\in N$)。 + +解:看到这个公式,虽然不知道如何利用递推公式得到这个反常积分的值,但是看到这个式子是由两个部分的乘积构成,所以尝试使用分部积分来计算看看: + +$I_n=-\int_0^{+\infty}x^n\,\textrm{d}(e^{-x})=[-x^ne^{-x}]_0^{+\infty}+\int_0^{+\infty}nx^{n-1}e^{-x}\,\textrm{d}x=-\lim\limits_{x\to+\infty}x^ne^{-x}+n\int_0^{+\infty}x^{n-1}e^{-x}\,\textrm{d}x=n\int_0^{+\infty}x^{n-1}e^{-x}\,\textrm{d}x=nI_{n-1}$。 + +且$I_0=\int_0^{+\infty}e^{-x}\,\textrm{d}x=[-e^{-x}]_0^{+\infty}=1$。 + +$\therefore I_n=n!$。 + \section{一元函数积分应用} \subsection{几何应用}