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Next: $B$3$NJ8=q$K$D$$$F(B... (PDF ե: integral.pdf)

$BJ?@.(B 13 $BG/(B 5 $B7n(B 23 $BF|(B
$BDj@QJ,$K$h$kCV49@QJ,!"ItJ,@QJ,$N8x<0$K$D$$$F(B
$B?73c9)2JBg3X(B $B>pJsEE;R9)3X2J(B $BC]LnLP<#(B

$B652J=q(B 86 $BJG$NDjM}(B 11.3 $B$K
$BDjM}(B 1

  1. $x=\phi(t)$ $B$OHyJ,2DG=$G(B $\phi(\alpha)=a$, $\phi(\beta)=b$ $B$G$"$k$H$-!"(B

    \begin{displaymath}
\int_a^b f(x) dx = \int_\alpha^\beta f(\phi(t))\phi'(t) dt
\end{displaymath}

  2. $f=f(x)$, $g=g(x)$ $B$,HyJ,2DG=$G$"$k$H$-!"(B

    \begin{displaymath}
\int_a^b fg' dx = [fg]_a^b - \int_a^b f'g dx
\end{displaymath}


$B$7$+$7!"NcG/$3$N8x<0$N1?MQ$K4X$9$k4V0c$$$d=q$-J}$,Hs>o$KB?$$!#(B $BBeI=E*$J$b$N$r$"$2$k!#(B

$BKhG/?'!9$J9)IW$r$7$F$=$N8x<0$r4V0c$$$J$/1?MQ$G$-$k$h$&$K65$($F$-$?$,!"(B $B
  • $BDj@QJ,$r(B

    \begin{displaymath}
\int_a^b f(x) dx = \left[\int f(x) dx\right]_{x=a}^{x=b}
\end{displaymath}

    $B$H9M$($k(B ($BCV49@QJ,$G$J$1$l$P(B ``$x=$'' $B$H$O=q$/I,MW$O$J$$(B)$B!#(B
  • $BCV49@QJ,!"ItJ,@QJ,$O$=$N(B $[ ]$ $B$NCf$G(B$BITDj@QJ,$H$7$F(B $B9T$&!#(B
  • $B$3$l$rMQ$$$?Nc$r>R2p$9$k!#(B


    $BNc(B 4


    \begin{displaymath}
I = \int_0^{\pi/2} 3\sin^2 x\cos x dx
= \left[\int 3\sin^2 x\cos x dx\right]_{x=0}^{x=\pi/2}
\end{displaymath}

    $B$H$7$^$:!"(B$[ ]$ $BFb$r7W;;$9$k!#(B$u=\sin x$ $B$H$9$k$H(B

    \begin{displaymath}
\frac{du}{dx}=\frac{d}{dx}\sin x = \cos x \mbox{ $B$h$j(B } du = \cos x dx
\end{displaymath}

    $B$H$J$k$N$G(B

    \begin{displaymath}
I = \left[\int 3u^2 du\right]_{x=0}^{x=\pi/2}
= \left[u^3+C\right]_{x=0}^{x=\pi/2}
\end{displaymath}

    $B$H$J$k$,!"$3$3$+$i@h$O(B

    \begin{displaymath}
I = \left[u^3+C\right]_{u=0}^{u=1} = (1+C)-(0-C) = 1
\end{displaymath}

    $B$G$b(B

    \begin{displaymath}
I = \left[\sin^3 x+C\right]_{x=0}^{x=\pi/2} = \sin^3\frac{\pi}{2} -\sin^3 0=1
\end{displaymath}

    $B$G$b$I$A$i$G$b$$$$$@$m$&!#MW$9$k$K!"BeF~$9$kJQ?t$,$"$C$F$$$l$P$$$$!#(B $B$J$*!"(B$[ ]$ $BFb$N(B $C$ $B$O$D$1$F$b$D$1$J$/$F$b$$$$!#(B



    $BNc(B 5


    \begin{displaymath}
I = \int_0^{\pi/2} x\cos x dx =\left[ \int x\cos x dx \right]_0^{\pi/2}
\end{displaymath}

    $B$H$7$F!"(B$[ ]$ $BFb$rItJ,@QJ,$9$k!#(B

    \begin{eqnarray*}I & = & \left[\int x (\sin x)' dx\right]_0^{\pi/2} \\
& = & \...
...} + C\right)
-(0\sin 0 + \cos 0 + C)\\
& = & \frac{\pi}{2}-1
\end{eqnarray*}



    $[ ]$ $B$NCf$N(B $C$ $B$O$D$1$F$b$D$1$J$/$F$b$$$$!#(B





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    Next: $B$3$NJ8=q$K$D$$$F(B...
    Shigeharu TAKENO
    2001$BG/(B 8$B7n(B 9$BF|(B