치환적분

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개요

치환적분법은 다음과 같다.

복잡한 함수를 적분할 때 쓰인다.

부정적분

$x=g\left(t\right)$

$\int f\left(x\right)dx=\int f\left\{g\left(t\right)\right\}\frac{dx}{dt}dt$
$\int f\left(x\right)dx=\int f\left\{g\left(t\right)\right\}{g}^{\prime }\left(t\right)dt$

예제 1

$\int {(ax+b)}^{n}dx$
$ax+b=t$
$ax=t-b$
$x=\frac{t-b}{a}$
$\frac{dx}{dt}=\frac{1}{a}$
$\int t^n\frac{1}{a}dt$
$\frac{1}{a}\int t^{n}dt$
$\frac{1}{a}\times \frac{t^{n+1}}{n+1}+C$
$\frac{t^{n+1}}{an+a}+C$
$\frac{{(ax+b)}^{n+1}}{an+a}+C$

예제 2

$\int \sin \left(3x\right)dx$
$3x=t$
$x=\frac{1}{3}t$
$\frac{dx}{dt}=\frac{1}{3}$
$\int \frac{1}{3}\sin tdt$
$\frac{1}{3}\int \sin tdt$
$\frac{1}{3}\times (-1)\cos t+C$
$-\frac{1}{3}\times \cos \left(3x\right)+C$
$-\frac{\cos \left(3x\right)}{3}+C$

예제 3

$\int x{e}^{-x^2}dx$
$-x^2=t$
$x^2=-t$
$x=\sqrt{-t}$
$\frac{dx}{dt}=-\frac{1}{2\sqrt{-t}}$
$\int \frac{-\sqrt{-t}{e}^t}{2\sqrt{-t}}dt$
$\int \frac{-e^t}{2}dt$
$-\frac{1}{2}\int e^{t}dt$
$-\frac{e^t}{2}+C$
$-\frac{e^{-x^2}}{2}+C$

예제 4

$\int \frac{f^{\prime }\left(x\right)}{f\left(x\right)}dx$
$f\left(x\right)=t$
$x=f^{-1}\left(t\right)dt$
$\frac{dx}{dt}=\frac{1}{f^{\prime }\left\{f^{-1}\left(t\right)\right\}}$
$\int \frac{f^{\prime }\left\{f^{-1}\left(t\right)\right\}}{t{f}^{\prime }\left\{f^{-1}\left(t\right)\right\}}dt$
$\int \frac{1}{t}dt$
$\ln \left|t\right|+C$
$\ln \left|f\left(x\right)\right|+C$

예제 5

$\int \tan xdx$
$\int \frac{\sin x}{\cos x}dx$
$\int \frac{-(\cos x)}{\cos x}dx$
$-\int \frac{(\cos x)}{\cos x}dx$
$-\ln |\cos x|+C$

예제 6

$\int \frac{\sin (\ln x)}{x}dx$
$\ln x=t$
$x=e^t=g\left(t\right)=g^{\prime }\left(t\right)$
$\int \frac{e^t\sin t}{e^t}dt$
$\int \sin tdt=-\cos t+C$
$-\cos (\ln \left|x\right|)+C$

예제 7

$\int a^{bx+c}dx$
$bx+c=t$
$bx=t-c$
$x=\frac{t-c}{b}$
$\frac{1}{b}\int a^{t}dt$
$\frac{a^t}{b\ln a}+C$
$\frac{a^{bx+c}}{b\ln a}+C$

예제 8

$\int {\cos }^{5}xdx$
$\int {\cos }^{3}x(1-{\sin }^{2}x)dx$
$\sin x=t$
$x=\arcsin t$
$\frac{dx}{dt}=\frac{1}{\sqrt{1-t^2}}$
$\int \frac{{\{\cos (\arcsin t)\}}^3(1-t^2)}{\sqrt{1-t^2}}dt$
${\cos }^2(\arcsin t)+t^2=1$
${\cos }^2(\arcsin t)=1-t^2$
$\arcsin t$의 범위는 $[-\frac{\pi }{2},\frac{\pi }{2}]$이고 이 범위에서 $y=\cos x$는
0보다 크다. 따라서,
$\cos (\arcsin t)=\sqrt{1-t^2}$
$\int \frac{{\left\{\sqrt{1-t^2}\right\}}^3(1-t^2)}{\sqrt{1-t^2}}dt$
$\int \frac{{\left\{\sqrt{1-t^2}\right\}}^2\sqrt{1-t^2}(1-t^2)}{\sqrt{1-t^2}}dt$
$\int {(1-t^2)}^{2}dt$
$\int (1-2t^2+t^4)dt$
$\int 1dt+\int (-2)t^{2}dt+\int t^{4}dt$
$\int 1dt-2\int t^{2}dt+\int t^{4}dt$
$t-\frac{2t^3}{3}+\frac{t^5}{5}+C$
$\frac{15t-10t^3+3t^5}{15}+C$
$\frac{15\sin x-10{\sin }^{3}x+3{\sin }^{5}x}{15}+C$

예제 9

$\int \frac{x}{\sqrt{1-x^2}}dx$
$x=\sin t=g\left(t\right)$
$g^{\prime }\left(t\right)=\cos t$
$\int \frac{\sin t\cos t}{\sqrt{1-{\sin }^{2}t}}dt$
${\sin }^{2}t+{\cos }^{2}t=1$
${\cos }^{2}t=1-{\sin }^{2}t$
$\int \frac{\sin t\cos t}{\sqrt{{\cos }^{2}t}}dt$
$\int \frac{\sin t\cos t}{\cos t}dt$
$\int \sin tdt=-\cos t+C$
$\arcsin x=t$
$-\cos (\arcsin x)+C$
그런데 예제 8에서도 입증됐듯이 $\cos (\arcsin x)$은 $\sqrt{1-x^2}$이므로.
$-\sqrt{1-x^2}+C$
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예제 10

• 1.$\int \frac{1}{x\sqrt{\ln x+1}}dx$
• 2.$\ln x+1=t$
  
• 3.$\ln x=t-1$
  
• 4.$x=e^{t-1}$
  
• 5.$\frac{dx}{dt}=e^{t-1}$
  
• 6.$\int \frac{e^{t-1}}{e^{t-1}\sqrt{t}}dt$
  
• 7.$\int \frac{1}{\sqrt{t}}dt$
  
• 8.$\int t^{-\frac{1}{2}}dt$
  
• 9.$\frac{t^{\frac{1}{2}}}{-\frac{1}{2}+1}+C$
  
• 10.$\frac{t^{\frac{1}{2}}}{\frac{1}{2}}+C$
  
• 11.$\frac{2\sqrt{t}}{1}+C$
  
• 12.$2\sqrt{t}+C$
  
• 13.$2\sqrt{\ln x+1}+C$
  

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예제 11

• 1.$\int \frac{1}{1-e^x}dx$
  
• 2.$1-e^x=t$
  
• 3.$-e^x=t-1$
  
• 4.$e^x=1-t$
  
• 5.$x=\ln |1-t|$
  
• 6.$\frac{dx}{dt}=\frac{-1}{1-t}=\frac{1}{t-1}$
  
• 7.$\int \frac{1}{t(t-1)}dt$
  
• 8.$\frac{1}{t(t-1)}=\frac{A}{t}+\frac{B}{t-1}$
  
• 9.$\frac{1}{t(t-1)}=\frac{A(t-1)+Bt}{t(t-1)}$
  
• 10.$\frac{1}{t(t-1)}=\frac{At-A+Bt}{t(t-1)}$
  
• 11.$\frac{1}{t(t-1)}=\frac{(A+B)t-A}{t(t-1)}$
  
• 12.$\{\begin{array}{cc}& A+B=0\\ & -A=1\end{array}$
  
• 13.$A=-1$
  
• 14.$-1+B=0$
  
• 15.$B=1$
  
• 16.$\int \frac{1}{t(t-1)}dt=\int (\frac{1}{t-1}-\frac{1}{t})dt$
  
• 17.$\int \frac{1}{t-1}dt+\int (-1)\frac{1}{t}dt$
  
• 18.$\int \frac{1}{t-1}dt-\int \frac{1}{t}dt$
  
• 19.$\ln |t-1|-\ln \left|t\right|+C$
  
• 20.$\ln \left|\frac{t-1}{t}\right|+C$
  
• 21.$\ln \left|\frac{1-e^x-1}{1-e^x}\right|+C$
  
• 22.$\ln \left|\frac{-e^x}{1-e^x}\right|+C$
  
• 23.$\ln \left|\frac{e^x}{e^x-1}\right|+C$
  

예제 12

• 1.$\int \frac{\ln x}{x}dx$
  
• 2.$\ln x=t$
  
• 3.$x=e^t$
  
• 4.$\frac{dx}{dt}=e^t$
  
• 5.$\int \frac{t{e}^t}{e^t}dt$
  
• 6.$\int tdt$
  
• 7.$\frac{t^2}{2}+C$
  
• 8.$\frac{{(\ln x)}^2}{2}+C$
  

예제 13

• 1.$\int \frac{1}{\sin x}dx$
  
• 2.$\int \frac{\sin x}{{\sin }^{2}x}dx$
  
• 3.$\int \frac{\sin x}{1-{\cos }^{2}x}dx$
  
• 4.$\cos x=t$
  
• 5.$x=\arccos t$
  
• 6.$f\left\{f^{-1}\left(x\right)\right\}=x$
  
• 7.$y=f\left(u\right),u=f^{-1}\left(x\right)$
  
• 8.$\frac{dy}{du}\frac{du}{dx}=f^{\prime }\left(u\right)\left\{f^{-1}\left(x\right)\right\}$
  
• 9.$f^{\prime }\left\{f^{-1}\left(x\right)\right\}\left\{f^{-1}\left(x\right)\right\}=1$
  
• 10.$\left\{f^{-1}\left(x\right)\right\}=\frac{1}{f^{\prime }\left\{f^{-1}\left(x\right)\right\}}$
  
• 11.$f\left(x\right)=\cos x$이렇게 두면 역함수의 미분법은
  
• 12.$f^{\prime }\left(x\right)=-\sin x$
  
• 13.$\frac{1}{-\sin (\arccos x)}$
  
• 14.${\sin }^{2}x+{\cos }^{2}x=1$
  
• 15.${\sin }^2(\arccos x)+x^2=1$
  
• 16.${\sin }^2(\arccos x)=1-x^2$
  
• 17.우선 $\arccos x$의 범위는 $[0,\pi ]$이고 이 범위에서 $\sin x$는 0보다 크다.따라서,
  
• 18.$\sin (\arccos x)=\sqrt{1-x^2}$
  
• 19.$(\arccos x)=\frac{1}{-\sqrt{1-x^2}}$
  
• 20.$(\arccos x)=\frac{-1}{\sqrt{1-x^2}}$
  
• 21.$(\arccos x)=-\frac{1}{\sqrt{1-x^2}}$
  
• 22.$\frac{dx}{dt}=-\frac{1}{\sqrt{1-t^2}}$
  
• 23.$\int \frac{\sqrt{1-t^2}}{(1-t^2)\sqrt{1-t^2}}dt$
  
• 24.$\int \frac{\bcancel{\sqrt{1-t^2}}}{(1-t^2)\bcancel{\sqrt{1-t^2}}}dt$
  
• 25.$\int \frac{-1}{1-t^2}dt$
  
• 26.$\frac{1}{1-t^2}=\frac{A}{1+t}+\frac{B}{1-t}$
  
• 27.$\frac{1}{1-t^2}=\frac{A(1-t)+B(1+t)}{1-t^2}$
  
• 28.$\frac{1}{1-t^2}=\frac{A-At+B+Bt}{1-t^2}$
  
• 29.$\frac{1}{1-t^2}=\frac{(B-A)t+(A+B)}{1-t^2}$
  
• 30.$\{\begin{array}{cc}& B-A=0\\ & A+B=1\end{array}$
  
• 31.$B=A$
  
• 32.$A+A=1$
  
• 32.$2A=1$
  
• 33.$A=\frac{1}{2}$
  
• 34.$\{\begin{array}{cc}& A=\frac{1}{2}\\ & B=\frac{1}{2}\end{array}$
  
• 35.$-\int (\frac{\frac{1}{2}}{1+t}+\frac{\frac{1}{2}}{1-t})dt$
  
• 36.$-\frac{1}{2}\int (\frac{1}{1+t}+\frac{1}{1-t})dt$
  
• 37.$-\frac{1}{2}\int \frac{1}{1+t}dt+\frac{1}{2}\int \frac{1}{1-t}dt$
  
• 38.$-\frac{1}{2}\int \frac{(1+t)}{1+t}dt+\frac{1}{2}\int \frac{-(1-t)}{1-t}dt$
  
• 39.$-\frac{1}{2}\int \frac{(1+t)}{1+t}dt-\frac{1}{2}\int \frac{(1-t)}{1-t}dt$
  
• 40.$-\frac{1}{2}\ln |1+t|-\frac{1}{2}\ln |1-t|+C$
  
• 41.$-\frac{1}{2}(\ln |1+t|-\ln |1-t|)+C$
  
• 42.$-\frac{1}{2}\ln \left|\frac{1+t}{1-t}\right|+C$
  
• 43.$-\frac{1}{2}\ln \left|\frac{1+\cos x}{1-\cos x}\right|+C$
  

예제 14

• 1.$\int \sin (ax+b)dx$
  
• 2.$ax+b=t$
  
• 3.$ax=t-b$
  
• 4.$x=\frac{t-b}{a}=g\left(t\right)$
  
• 5.$g^{\prime }\left(t\right)=\frac{1}{a}$
  
• 6.$\frac{1}{a}\int \sin tdt$
  
• 7.$\frac{-\cos t}{a}+C$
  
• 8.$\frac{-\cos (ax+b)}{a}+C$
  

예제 15

\( \require{cancel} \displaystyle \int \frac{1}{x\ln x}dx\\ \displaystyle \ln x=t \\ \displaystyle x=e^{t}=g\left ( t \right ) \\ \displaystyle g'\left ( t \right )=e^{t} \\ \displaystyle \int \frac{e^{t}}{t e^{t}}dt \\ \displaystyle \int \frac{\cancel{e^{t}}}{t \cancel{e^{t}}}dt\\ \displaystyle \int \frac{1}{t}dt\\ \displaystyle \ln\left | t \right |+C \\ \displaystyle \ln\left | \ln x \right |+C\\ \)

정적분

닫힌 구간 [a,b]에서 연속인 함수 f(x)에 대하여 미분가능한 함수

$x=g\left(t\right)$

의 도함수 $g^{\prime }\left(t\right)$가 닫힌 구간 $[\alpha ,\beta ]$에서 연속이고,
$a=g\left(\alpha \right),b=g\left(\beta \right)$이면,
$\underset{a}{\overset{b}{\int }}f\left(x\right)dx=\underset{\alpha }{\overset{\beta }{\int }}f\left\{g\left(t\right)\right\}{g}^{\prime }\left(t\right)dt$

예제 1

$\underset{0}{\overset{a}{\int }}\sqrt{a^2-x^2}dx$
$x=a\sin t=g\left(t\right)$
$0=a\sin \alpha$
$sin\alpha =0$
$\alpha =0$
$a=a\sin \beta$
$1=\sin \beta$
$\frac{\pi }{2}=\beta$
$g^{\prime }\left(t\right)=a\cos t$
$\underset{0}{\overset{\frac{\pi }{2}}{\int }}\sqrt{a^2-a^2{\cos }^{2}t}a\cos tdt$
$\underset{0}{\overset{\frac{\pi }{2}}{\int }}\sqrt{a^2-a^2{\sin }^{2}t}a\cos tdt$
$\underset{0}{\overset{\frac{\pi }{2}}{\int }}\sqrt{a^2(1-{\sin }^{2}t)}a\cos tdt$
$\underset{0}{\overset{\frac{\pi }{2}}{\int }}\sqrt{a^2{\cos }^{2}t}a\cos tdt$
$\underset{0}{\overset{\frac{\pi }{2}}{\int }}a\cos t\times a\cos tdt$
$a^2\underset{0}{\overset{\frac{\pi }{2}}{\int }}{\cos }^{2}tdt$
$a^2\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{\cos \left(2t\right)+1}{2}dt$
$a^2(\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{\cos \left(2t\right)}{2}dt+\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{1}{2}dt)$
$a^2(\frac{1}{2}\underset{0}{\overset{\frac{\pi }{2}}{\int }}\cos \left(2t\right)dt+\frac{1}{2}\underset{0}{\overset{\frac{\pi }{2}}{\int }}dt)$
$\frac{a^2}{2}(\underset{0}{\overset{\frac{\pi }{2}}{\int }}\cos \left(2t\right)dt+\underset{0}{\overset{\frac{\pi }{2}}{\int }}dt)$
$2t=k$
$t=\frac{1}{2}k$
$t=0,k=0$
$t=\frac{\pi }{2},k=\pi$
$\frac{a^2}{2}(\frac{1}{2}\underset{0}{\overset{\pi }{\int }}\cos \left(k\right)dk+\underset{0}{\overset{\frac{\pi }{2}}{\int }}dt)$
$\frac{a^2}{2}(\frac{1}{2}{[\sin k]}_0^{\pi }+{\left[t\right]}_0^{\frac{\pi }{2}})$
$\frac{a^2}{2}\{\frac{1}{2}(\sin \pi -\sin 0)+\frac{\pi }{2}\}$
<math>\frac{a^{2}}{2}\times\frac{\pi}{2}=\frac{\pi a^{2}}{4}