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zN9ddHXaC9P`TG^e>nDHfx>Ph*d^4MR8S?$o37U&xqfL#<=y&0C=p6;#D3^5~8Bcg= m9WM7RO^z~ox4;qq0@dt9YMe3%9yB8d10xhEsi?dd)PDik#(lZ~ 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 8ebb573..172b6a2 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 @@ -97,7 +97,7 @@ $=\dfrac{1}{4}\int(\sin2x+\sin4x)\textrm{d}x-\dfrac{1}{12}\sin^23x=-\dfrac{1}{16 \paragraph{三角等式} \leavevmode \medskip -主要用于$\sec^2-1=\tan^2x$,当出现$\tan^2$、$\tan^3$等与$\sec x$在一起作为乘积时可以考虑拆分换元。 +主要用于$\sec^2-1=\tan^2x$,当出现$\tan^2$、$\tan^3$等与$\sec x$在一起作为乘积时可以考虑拆分换元。同时$\sin^2x+\cos^2=1$,$\csc^2x-\cos^2x=1$。 \textbf{例题:}$\displaystyle{\int\tan^3x\sec x\,\textrm{d}x}$。 @@ -105,9 +105,19 @@ $=\dfrac{1}{4}\int(\sin2x+\sin4x)\textrm{d}x-\dfrac{1}{12}\sin^23x=-\dfrac{1}{16 $=\dfrac{1}{3}\sec^3x-\sec x+C$。 -需要利用到有理积分的高阶多项式分配与低阶多项式因式分解。 +有时需要利用到有理积分的高阶多项式分配与低阶多项式因式分解。 -有时还可以用二倍角公式。\medskip +\textbf{例题:}求$\displaystyle{\int\dfrac{\textrm{d}x}{(2+\cos x)\sin x}}$。\medskip + +解:将所有的$\sin x$都变成$\cos x$。所以分子分母都乘上$\sin x$。 + +$=\displaystyle{\int\dfrac{\textrm{d}(\cos x)}{(2+\cos x)(\cos^2x-1)}}$,令$\cos x=u$:$=\displaystyle{\int\dfrac{\textrm{d}u}{(2+u)(u^2-1)}}$。 + +利用低阶多项式因式分解:$=\displaystyle{\int\left[\dfrac{1}{6(u-1)}-\dfrac{1}{2(u+1)}+\dfrac{1}{3(u+2)}\right]\textrm{d}u}$ + +$=\dfrac{1}{6}\vert u-1\vert-\dfrac{1}{2}\ln\vert u+1\vert+\dfrac{1}{3}\ln\vert u+2\vert+C=\dfrac{1}{6}\ln(1-\cos x)-\dfrac{1}{2}\ln(1+\cos x)+\dfrac{1}{3}\ln(2+\cos x)+C$。 + +有时还可以用二倍角公式。特别是遇到$1+\cos x$等。\medskip \textbf{例题:}求$\displaystyle{\int\dfrac{\sqrt{1+\cos x}}{\sin x}\textrm{d}x}$。 @@ -123,7 +133,15 @@ $=\displaystyle{\int\dfrac{\sqrt{2}\left\vert\cos\dfrac{x}{2}\right\vert}{2\sin\ 所以当$\cos\dfrac{x}{2}>0$时乘积正负号不变,$=\ln\left(\left\vert\csc\dfrac{x}{2}\right\vert-\left\vert\cot\dfrac{x}{2}\right\vert\right)$。 -当$\cos\dfrac{x}{2}<0$时乘积正负号相反,$=\ln\left(\left\vert\csc\dfrac{x}{2}\right\vert+\left\vert\cot\dfrac{x}{2}\right\vert\right)$。 +当$\cos\dfrac{x}{2}<0$时乘积正负号相反,提出负号,$=\ln\left(\left\vert\csc\dfrac{x}{2}\right\vert+\left\vert\cot\dfrac{x}{2}\right\vert\right)$。 + +又$\csc^2\dfrac{x}{2}-\cot^2\dfrac{x}{2}=1$,$\therefore\ln\left(\left(\left\vert\csc\dfrac{x}{2}\right\vert+\left\vert\cot\dfrac{x}{2}\right\vert\right)\cdot\left(\left\vert\csc\dfrac{x}{2}\right\vert-\left\vert\cot\dfrac{x}{2}\right\vert\right)\right)=\ln\left(\csc^2\dfrac{x}{2}-\cot^2\dfrac{x}{2}\right)=\ln1=0$,$\therefore\ln\left(\left\vert\csc\dfrac{x}{2}\right\vert+\left\vert\cot\dfrac{x}{2}\right\vert\right)+\ln\left(\left\vert\csc\dfrac{x}{2}\right\vert-\left\vert\cot\dfrac{x}{2}\right\vert\right)=0$,$\ln\left(\left\vert\csc\dfrac{x}{2}\right\vert+\left\vert\cot\dfrac{x}{2}\right\vert\right)=-\ln\left(\left\vert\csc\dfrac{x}{2}\right\vert-\left\vert\cot\dfrac{x}{2}\right\vert\right)$。 + +当$\cos\dfrac{x}{2}>0$,$=\sqrt{2}\ln\left\vert\csc\dfrac{x}{2}-\cot\dfrac{x}{2}\right\vert+C=\sqrt{2}\ln\left(\left\vert\csc\dfrac{x}{2}\right\vert-\left\vert\cot\dfrac{x}{2}\right\vert\right)+C$。 + +当$\cos\dfrac{x}{2}<0$,$=-\sqrt{2}\ln\left\vert\csc\dfrac{x}{2}-\cot\dfrac{x}{2}\right\vert+C=\sqrt{2}\ln\left(\left\vert\csc\dfrac{x}{2}\right\vert-\left\vert\cot\dfrac{x}{2}\right\vert\right)+C$。 + +综上,$\displaystyle{\int\dfrac{\sqrt{1+\cos x}}{\sin x}\textrm{d}x}=\sqrt{2}\ln\left(\left\vert\csc\dfrac{x}{2}\right\vert-\left\vert\cot\dfrac{x}{2}\right\vert\right)+C$。 \paragraph{反三角换元} \leavevmode \medskip @@ -677,8 +695,20 @@ $\therefore\displaystyle{\int_0^\frac{\pi}{2}\dfrac{\sin x}{\cos x+\sin x}\textr \subsubsection{换元积分} +换元积分法基本上跟不定积分一样。 + +\textbf{例题:}求$\displaystyle{\int_0^2\dfrac{x\,\textrm{d}x}{(x^2-2x+2)^2}}$。\medskip + +解:从题目来看都需要使用换元积分法,因为分母比较复杂。一种思路直接令$x^2-2x+2=u$,但是这样$\textrm{d}x$就解不出来。所以尝试将分母配方:$x^2-2x+2=(x-1)^2+1$,看到$x^2+1$的形式就想到了$\tan^2u+1=\sec^2u$,所以令$x-1=\tan u$,即$x=\tan u+1$,$\textrm{d}x=\sec^2u\,\textrm{d}u$,$(x^2-2x+2)^2=\sec^4u$。当$x=0$,$u=-\dfrac{\pi}{4}$,当$x=2$,$u=\dfrac{\pi}{4}$。解得$=\displaystyle{\int_{-\frac{\pi}{4}}^\frac{\pi}{4}\dfrac{(1+\tan u)\textrm{d}u}{\sec^2u}}$ + +$=\int_{-\frac{\pi}{4}}^\frac{\pi}{4}\cos^2u\,\textrm{d}u+\int_{-\frac{\pi}{4}}^\frac{\pi}{4}\cos^2u\tan u\,\textrm{d}u=\int_{-\frac{\pi}{4}}^\frac{\pi}{4}\cos^2u\,\textrm{d}u+\int_{-\frac{\pi}{4}}^\frac{\pi}{4}\sin u\cos u\,\textrm{d}u=\int_{-\frac{\pi}{4}}^\frac{\pi}{4}\cos^2u\,\textrm{d}u=2\int_0^\frac{\pi}{4}\cos^2u\,\textrm{d}u=2\int_0^\frac{\pi}{4}(1+\cos2u)\,\textrm{d}u=\dfrac{\pi}{4}+\dfrac{1}{2}$。 + \subsubsection{分部积分} +可以使用普通的分部积分方式,也可以使用分部积分推广公式。 + + + \subsection{变限积分} \subsubsection{极限}