부분적분법

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부정적분

부분적분법은 다음과 같다.

두 함수 f(x),g(x)가 미분가능할 때 다음이 성립한다.

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

예제 1

$\int x{e}^{x}dx$
$f\left(x\right)=x,g^{\prime }\left(x\right)=e^x$
$f^{\prime }\left(x\right)=1,g\left(x\right)=e^x+C$
$x(e^x+C)-\int (e^x+C)dx$
$x{e}^x+Cx-(\int e^{x}dx+\int Cdx)$
$x{e}^x+Cx-\int e^{x}dx-C\int 1dx$
$x{e}^x+Cx-e^x-Cx+D$
$x{e}^x+\bcancel{Cx}-e^x\bcancel{-Cx}+D$
$x{e}^x-e^x+D$
$e^x(x-1)+D$

예제 2

$\int x\cos xdx$
$f\left(x\right)=x,g^{\prime }\left(x\right)=\cos x$
$f^{\prime }\left(x\right)=1,g\left(x\right)=\sin x$
$x\sin x-\int \sin xdx$
$x\sin x+\cos x+C$

LIATE 기법

LIATE는 각각 대수함수,역삼각함수,다항함수,삼각함수,지수함수를 말하는데 여기서 앞에 있는 것을 $f\left(x\right)$에 넣고
나머지 함수를 $g^{\prime }\left(x\right)$에 넣으면 좀 더 수월하게 부정적분을 구할 수 있다.

예제 3

$\int e^x\sin xdx$
$f\left(x\right)=\sin x,g^{\prime }\left(x\right)=e^x$
$e^x\sin x-\int e^x\cos xdx$
$e^x\sin x-\int e^x\cos xdx$
$\int e^x\cos xdx$
$f_1\left(x\right)=\cos x,g_{1^{\prime }}\left(x\right)=e^x$
$f_{1^{\prime }}\left(x\right)=-\sin x,g_1\left(x\right)=e^x$
$e^x\cos x+\int e^x\sin xdx$
$\int e^x\sin xdx=e^x\sin x-e^x\cos x-\int e^x\sin x$
$2\int e^x\sin xdx=e^x(\sin x-\cos x)+C$
$\int e^x\sin xdx=\frac{e^x(\sin x-\cos x)}{2}+C$

예제 4

$\int e^x\cos xdx$
$f\left(x\right)=e^x,g^{\prime }\left(x\right)=\cos x$
$f^{\prime }\left(x\right)=e^x,g\left(x\right)=\sin x$
$e^x\sin x-\int e^x\sin xdx$
$e^x\sin x-\left\{\frac{e^x(\sin x-\cos x)}{2}\right\}+C$
$\frac{2e^x\sin x-e^x\sin x+e^x\cos x}{2}+C$
$\frac{e^x\sin x+e^x\cos x}{2}+C$
$\frac{e^x(\sin x+\cos x)}{2}+C$

예제 5

$\int \cos x\ln |\sin x|dx$
$f\left(x\right)=\ln |\sin x|,g^{\prime }\left(x\right)=\cos x$
$f^{\prime }\left(x\right)=\frac{\cos x}{\sin x},g\left(x\right)=\sin x$
$\sin \left(x\right)\ln |\sin x|-\int \cos xdx$
$\sin \left(x\right)\ln |\sin x|-\sin x+C$
$\sin x(\ln |\sin x|-1)+C$

예제 6

• 1.$\int x\sin xdx$
  
• 2.$f\left(x\right)=x,g^{\prime }\left(x\right)=\sin x$
  
• 3.$f^{\prime }\left(x\right)=1,g\left(x\right)=-\cos x$
  
• 4.$-x\cos x+\int \cos xdx$
  
• 5.$-x\cos x+\sin x+C$
  

예제 7

• 1.$\int \ln xdx$
  
• 2.$f\left(x\right)=\ln x,g^{\prime }\left(x\right)=1$
  
• 3.$f^{\prime }\left(x\right)=\frac{1}{x},g\left(x\right)=x$
  
• 4.$x\ln x-\int 1dx$
  
• 5.$x\ln x-x+C$
  
• 6.$x(\ln x-1)+C$
  

예제 8

• 1.$\int {\sin }^{n}xdx(n\ge 2,n\in \mathbb{N})$
  
• 2.$\int {\sin }^{n-1}x\sin xdx$
  
• 3.$f\left(x\right)={\sin }^{n-1}x,g^{\prime }\left(x\right)=\sin x$
  
• 4.일단 $f\left(x\right)$의 도함수를 구해 보자.${\displaystyle }$이렇게 두면.
  
• 5.$f\left(u\right)=u^{n-1},u=\sin x$
  
• 6.$\frac{dy}{du}\frac{du}{dx}=(n-1)u^{n-2}\cos x$
  
• 7.$\frac{dy}{du}\frac{du}{dx}=(n-1){(\sin x)}^{n-2}\cos x$
  
• 8.$f^{\prime }\left(x\right)=(n-1){(\sin x)}^{n-2}\cos x,g\left(x\right)=-\cos x$
  
• 9.$-{\sin }^{n-1}x\cos x-\int \left\{{(-\cos x)}^2(n-1){(\sin x)}^{n-2}\right\}dx$
  
• 10.$-{\sin }^{n-1}x\cos x+\int \left\{{(\cos x)}^2(n-1){(\sin x)}^{n-2}\right\}dx$
  
• 11.$-{\sin }^{n-1}x\cos x+(n-1)\int \left\{{(\cos x)}^2{(\sin x)}^{n-2}\right\}dx$
  
• 12.$-{\sin }^{n-1}x\cos x+(n-1)\int \left\{\left({\cos }^{2}x\right){(\sin x)}^{n-2}\right\}dx$
  
• 13.$-{\sin }^{n-1}x\cos x+(n-1)\int \left\{(1-{\sin }^{2}x){(\sin x)}^{n-2}\right\}dx$
  
• 14.$\int {\sin }^{n}xdx=-{\sin }^{n-1}x\cos x+(n-1)\int \left\{(1-{\sin }^{2}x){(\sin x)}^{n-2}\right\}dx$
  
• 15.$\int {\sin }^{n}xdx=-{\sin }^{n-1}x\cos x+(n-1)\int ({\sin }^{n-2}x-{\sin }^{n}x)dx$
  
• 16.$\int {\sin }^{n}xdx=-{\sin }^{n-1}x\cos x+(n-1)\int {\sin }^{n-2}xdx+(n-1)\int (-1){\sin }^{n}xdx$
  
• 17.$\int {\sin }^{n}xdx=-{\sin }^{n-1}x\cos x+(n-1)\int {\sin }^{n-2}xdx+(1-n)\int {\sin }^{n}xdx$
  
• 18.$\int {\sin }^{n}xdx+(n-1)\int {\sin }^{n}xdx=-{\sin }^{n-1}x\cos x+(n-1)\int {\sin }^{n-2}xdx$
  
• 19.$(n-1+1)\int {\sin }^{n}xdx=-{\sin }^{n-1}x\cos x+(n-1)\int {\sin }^{n-2}xdx$
  
• 20.$n\int {\sin }^{n}xdx=-{\sin }^{n-1}x\cos x+(n-1)\int {\sin }^{n-2}xdx$
  
• 21.$\therefore \int {\sin }^{n}xdx=\frac{-{\sin }^{n-1}x\cos x}{n}+\frac{n-1}{n}\int {\sin }^{n-2}xdx(n\ge 2,n\in \mathbb{N})$
  

예제 9

• 1.$\int {\cos }^{n}xdx$
  
• 2.$\int {\cos }^{n-1}x\cos xdx$
  
• 3.$f\left(x\right)={\cos }^{n-1}x,g^{\prime }\left(x\right)=\cos x$
  
• 4.$y=u^{n-1},u=\cos x$
  
• 5.$\frac{dy}{du}\frac{du}{dx}=(n-1)u^{n-2}(-1)\sin x$
  
• 6.$f^{\prime }\left(x\right)=-(n-1){(\cos x)}^{n-2}\sin x,g\left(x\right)=\sin x$
  
• 7.$\int {\cos }^{n}xdx={\cos }^{n-1}x\sin x-\int (-1)(n-1){\cos }^{n-2}x{\sin }^{2}xdx$
  
• 8.$\int {\cos }^{n}xdx={\cos }^{n-1}x\sin x+\int (n-1){\cos }^{n-2}x{\sin }^{2}xdx$
  
• 9.$\int {\cos }^{n}xdx={\cos }^{n-1}x\sin x+\int (n-1){\cos }^{n-2}x(1-{\cos }^{2}x)dx$
  
• 10.$\int {\cos }^{n}xdx={\cos }^{n-1}x\sin x+(n-1)\int {\cos }^{n-2}x(1-{\cos }^{2}x)dx$
  
• 11.$\int {\cos }^{n}xdx={\cos }^{n-1}x\sin x+(n-1)\int ({\cos }^{n-2}x-{\cos }^{n}x)dx$
  
• 12.$\int {\cos }^{n}xdx={\cos }^{n-1}x\sin x+(n-1)\int {\cos }^{n-2}xdx-(n-1)\int {\cos }^{n}xdx$
  
• 13.$\int {\cos }^{n}xdx={\cos }^{n-1}x\sin x+(n-1)\int {\cos }^{n-2}xdx+(1-n)\int {\cos }^{n}xdx$
  
• 14.$\int {\cos }^{n}xdx+(n-1)\int {\cos }^{n}xdx={\cos }^{n-1}x\sin x+(n-1)\int {\cos }^{n-2}xdx$
  
• 15.$(n-1+1)\int {\cos }^{n}xdx={\cos }^{n-1}x\sin x+(n-1)\int {\cos }^{n-2}xdx$
  
• 16.$n\int {\cos }^{n}xdx={\cos }^{n-1}x\sin x+(n-1)\int {\cos }^{n-2}xdx$
  
• 17.$\int {\cos }^{n}xdx=\frac{{\cos }^{n-1}x\sin x}{n}+\frac{n-1}{n}\int {\cos }^{n-2}xdx$
  

예제 10

• 1.$\int {\tan }^{n}x$
  
• 2.일단 다음과 같이 쪼갭니다.$\int {\tan }^{n-2}{\tan }^{2}xdx$
  
• 3.$f\left(x\right)={\tan }^{n-2}x,g^{\prime }\left(x\right)={\tan }^{2}x$
  
• 4.$\int {\tan }^{n-2}x({\sec }^{2}x-1)dx$을 구해보기 위해 다음과 같이 놓습니다.
  
• 5.$\int {\tan }^{n-2}x{\sec }^{2}xdx-\int {\tan }^{n-2}xdx=\int {\tan }^{n}xdx$
  
• 6.$f\left(x\right)={\tan }^{n-2}x,g^{\prime }\left(x\right)={\sec }^{2}x$
  
• 7.$f^{\prime }\left(x\right)=\frac{(n-2){\tan }^{n-1}x}{{\sin }^{2}x},g\left(x\right)=\tan x$
  
• 8.${\tan }^{n-1}x+(2-n)\int \frac{{\tan }^{n}x}{{\sin }^{2}x}dx$
  
• 9.${\tan }^{n-1}x+(2-n)\int {\tan }^{n}x{\csc }^{2}xdx$
  
• 10.${\tan }^{n-1}x+(2-n)\int {\tan }^{n}x(1+{\cot }^{2}x)dx$
  
• 11.${\tan }^{n-1}x+(2-n)\int ({\tan }^{n}x+{\tan }^{n-2}x)dx$
  
• 12.${\tan }^{n-1}x+(2-n)(\int {\tan }^{n}xdx+\int {\tan }^{n-2}xdx)$
  
• 13.${\tan }^{n-1}x+(2-n)\int {\tan }^{n}xdx+(2-n)\int {\tan }^{n-2}xdx$
  
• 14.${\tan }^{n-1}x+(2-n)\int {\tan }^{n}xdx+(2-n)\int {\tan }^{n-2}x-\int {\tan }^{n-2}xdx=\int {\tan }^{n}xdx$
  
• 15.${\tan }^{n-1}x+(2-n-1)\int {\tan }^{n-2}xdx=\int {\tan }^{n}xdx+(n-2)\int {\tan }^{n}xdx$
  
• 16.${\tan }^{n-1}x+(1-n)\int {\tan }^{n-2}xdx=(n-2+1)\int {\tan }^{n}xdx$
  
• 17.${\tan }^{n-1}x+(1-n)\int {\tan }^{n-2}xdx=(n-1)\int {\tan }^{n}xdx$
  
• 18.$\frac{{\tan }^{n-1}x}{n-1}+\frac{-(n-1)}{n-1}\int {\tan }^{n-2}xdx=\int {\tan }^{n}xdx$
  
• 19.$\frac{{\tan }^{n-1}x}{n-1}-\int {\tan }^{n-2}xdx=\int {\tan }^{n}xdx$
  

예제 11

\( \displaystyle \int \textrm{cosec}^{n}xdx\\ \displaystyle \int \textrm{cosec}^{n-2}x\textrm{cosec}^{2}xdx\\ \displaystyle y=u^{n-2},u=\textrm{cosec}x\\ \displaystyle \frac{dy}{du}=\left ( n-2 \right )u^{n-3},\frac{du}{dx}=-\cot x\textrm{cosec}x\\ \displaystyle \frac{dy}{du}\frac{du}{dx}=\left ( 2-n \right )\textrm{cosec}^{n-3}x\cot x\textrm{cosec}x\\ \displaystyle \begin{cases} & f\left ( x \right )=\textrm{cosec}^{n-2}x,g'\left ( x \right )= \textrm{cosec}^{2}x\\ & f'\left (x \right )= \left ( 2-n \right )\textrm{cosec}^{n-2}x\cot x,g\left ( x \right )=-\cot x\\ \end{cases}\\ \displaystyle \int \textrm{cosec}^{n}xdx=-\textrm{cosec}^{n-2}x\cot x+\left ( 2-n \right )\int \textrm{cosec}^{n-2}x\cot^{2} xdx\\ \displaystyle \int \textrm{cosec}^{n}xdx=-\textrm{cosec}^{n-2}x\cot x+\left ( 2-n \right )\int \textrm{cosec}^{n-2}x\cot^{2} xdx\\ \displaystyle \int \textrm{cosec}^{n}xdx=-\textrm{cosec}^{n-2}x\cot x+\left ( 2-n \right )\int \textrm{cosec}^{n-2}x\left ( -1+ \textrm{cosec}^{2}x \right )dx\\ \displaystyle \int \textrm{cosec}^{n}xdx=-\textrm{cosec}^{n-2}x\cot x+\left ( 2-n \right )\int \left ( -\textrm{cosec}^{n-2}x+\textrm{cosec}^{n}x \right )dx\\ \displaystyle \int \textrm{cosec}^{n}xdx=-\textrm{cosec}^{n-2}x\cot x+\left ( n-2 \right )\int \textrm{cosec}^{n-2}xdx+\left ( 2-n \right )\int \textrm{cosec}^{n}xdx\\ \displaystyle \int \textrm{cosec}^{n}xdx+\left ( n-2 \right )\int \textrm{cosec}^{n}xdx=-\textrm{cosec}^{n-2}x\cot x+\left ( n-2 \right )\int \textrm{cosec}^{n-2}xdx\\ \displaystyle \left ( n-2+1 \right )\int \textrm{cosec}^{n}xdx=-\textrm{cosec}^{n-2}x\cot x+\left ( n-2 \right )\int \textrm{cosec}^{n-2}xdx\\ \displaystyle \int \textrm{cosec}^{n}xdx=\frac{-\textrm{cosec}^{n-2}x\cot x}{n-1}+\frac{n-2}{n-1}\int \textrm{cosec}^{n-2}xdx \)

적분상수?

사실 적분상수를 붙여도 소거되므로 상관없다.

• 0.여기서 $G_1\left(x\right)+C$는 임의의 연속함수 $g_1\left(x\right)$의 부정적분이다.
  그리고 $f_1\left(x\right)$는 임의의 연속함수이고 $D$는 적분상수이다.
  
• 1.$f\left(x\right)=f_1\left(x\right),g^{\prime }\left(x\right)=g_1\left(x\right)$
  
• 2.$f^{\prime }\left(x\right)={f_1}^{\prime }\left(x\right),g\left(x\right)=G_1\left(x\right)+C$
  
• 3.$f_1\left(x\right)\{G_1\left(x\right)+C\}-\int f{\text{'}}_1\left(x\right)\{G_1\left(x\right)+C\}dx$
  
• 4.$f_1\left(x\right)\{G_1\left(x\right)+C\}-\int \{f{\text{'}}_1\left(x\right)G_1\left(x\right)+Cf{\text{'}}_1\left(x\right)\}dx$
  
• 5.$f_1\left(x\right)\{G_1\left(x\right)+C\}-(\int f{\text{'}}_1\left(x\right)G_1\left(x\right)dx+\int Cf{\text{'}}_1\left(x\right)dx)$
  
• 6.$f_1\left(x\right)\{G_1\left(x\right)+C\}-(\int f{\text{'}}_1\left(x\right)G_1\left(x\right)dx)-\int Cf{\text{'}}_1\left(x\right)dx$
  
• 7.$f_1\left(x\right)\{G_1\left(x\right)+C\}-\int f{\text{'}}_1\left(x\right)G_1\left(x\right)dx-C\int f{\text{'}}_1\left(x\right)dx$
  
• 8.$f_1\left(x\right)G_1\left(x\right)+C{f}_1\left(x\right)-\int f{\text{'}}_1\left(x\right)G_1\left(x\right)dx-C{f}_1\left(x\right)+D$
  
• 9.$f_1\left(x\right)G_1\left(x\right)+\cancel{C{f}_1\left(x\right)}-\int f{\text{'}}_1\left(x\right)G_1\left(x\right)dx\cancel{-C{f}_1\left(x\right)}+D$
• 10.$f_1\left(x\right)G_1\left(x\right)-\int f{\text{'}}_1\left(x\right)G_1\left(x\right)dx$
  

정적분

두 함수 f(x),g(x)가 미분가능하고 닫힌 구간 [a,b]에서 연속이면,

$\underset{a}{\overset{b}{\int }}f\left(x\right)g^{\prime }\left(x\right)dx={\left[f\left(x\right)g\left(x\right)\right]}_a^b-\underset{a}{\overset{b}{\int }}{f}^{\prime }\left(x\right)g\left(x\right)dx$