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{\large ���wI���K�𓚗�@��X��@2007�N10��30���o�蕪} 

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�S�� ���n����CTA �O�p �~
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\noindent{\bf [1]}
(1)	
$\displaystyle
\int \frac{x}{1+x+x^2+x^3} dx
=\int \frac{1}{2} \left( 
\frac{1}{1+x^2} +\frac{x}{1+x^2} 
-\frac{1}{1+x} \right) dx 
=\frac{1}{2} \arctan x+\frac{1}{4} 
\log (1+x^2 ) -\frac{1}{2} \log |1+x| +C$.

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(2)	
$x=a\sin y$ 
($-\pi /2 <y<\pi /2$) 
�Ƃ����Ēu���ϕ�����ƁA

$\displaystyle\int 
\frac{1}{\sqrt{a^2 -x^2 }}dx
=\int dy=\arcsin\frac{x}{a} +C$.

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(3)	
$\displaystyle
\int \frac {{\cos}^2 x}{(\cos x +x\sin x)^2} 
dx=-\int {\left( \frac{1}{\cos x+x\sin x} 
\right)}' \frac{\cos x}{x} dx$
������A�����ϕ��ɂ��
(�E��)$\displaystyle=
\frac{\sin x}{\cos x +x\sin x} +C$.

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\noindent{\bf [2]}  
$n\ge 2$�ɑ΂��āA
\begin{eqnarray*}
I_n &=&\int_{0}^{\pi /4} \left( \frac{1}{{\cos}^2 x}
-1 \right) {\tan}^{n-2} xdx\\
&=& \int_{0}^{\pi /4} (\tan x)' {\tan}^{n-2} xdx- I_{n-2}. 
\end{eqnarray*}
�E�ӑ�ꍀ�̐ϕ���$J_n$�Ƃ����ƁA
$J_n =1-(n-2)\int_{0}^{\pi /4} (\tan x)' 
{\tan}^{n-2} xdx=1-(n-2) J_n $
������$J_n =\frac{1}{n-1}$�B
�����$I_n =\frac{1}{n-1} - I_{n-2}$�B

�܂��A$I_0 =\pi /4$, $I_1 =\frac{1}{2} \log 2$�B
�������A�e$m$�ɑ΂���
\begin{eqnarray*}
I_{2m} &=&\frac{1}{2m-1} -\frac{1}{2m-3} +\cdots 
+(-1)^{m-1} +(-1)^m \frac{\pi}{4},\\
I_{2m+1} &=&\frac{1}{2m} -\frac{1}{2m-2} +\cdots 
+(-1)^{m-1} \frac{1}{2} +(-1)^m \frac{1}{2} \log 2.
\end{eqnarray*}

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\noindent{\bf [3]}
\begin{eqnarray*}
&{}&\int_{-1}^{1} \frac{h}{h^2 +x^2 } f(x)dx\\
&=&\int_{-1}^{1} \frac{h}{h^2 +x^2 } \{f(x)-f(0)\}dx 
+\int_{-1}^{1} \frac{h}{h^2 +x^2 } f(0)dx
\equiv I_1 +I_2 
\end{eqnarray*}
�Ƃ����B
$I_2 =2f(0) \arctan \frac{1}{h} \to\pi f(0)$ 
($h\to 0+$) �ł���B
$I_1$��$0$�Ɏ������鎖�������΂悢�B

����$f$��$[-1, 1]$��ŘA�����A���萔$M$
�����݂���$|f(x)|\le M$���݂����B�܂����_�ɂ�����
�A�������A�C�ӂ�$\e >0$�ɑ΂���$\d\in (0,1)$
�����݂��āA$|x|<\d$�Ȃ��$|f(x)-f(0)|<\e$���݂����B
���̂Ƃ��A
\begin{eqnarray*}
| I_1 |&\le& \int_{-1}^{1} \frac{h}{h^2 +x^2 } 
|f(x)-f(0)|dx\\
&\le& \int_{-1}^{-\d} \frac{h}{h^2 +x^2 } 
2Mdx+ 
\int_{-\d}^{\d} \frac{h}{h^2 +x^2 } 
\e dx+ 
\int_{\d}^{1} \frac{h}{h^2 +x^2 } 
2Mdx\\
&=&4M(\arctan \frac{1}{h} -\arctan \frac{\d}{h} )
+2\e \arctan \frac{\d}{h} \to \e\pi. \quad (h\to 
0+)
\end{eqnarray*}
$\e$�͔C�ӂ���������A
$\lim_{h\to 0+} I_1 =0$�B

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\noindent�E
[1] (2)�͐ϕ��萔�����点��
$-\arccos \frac{x}{a}$�ł������ł��B

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\noindent�E 
[3]��$I_1$�̔�ϕ��֐��́A$x$���Œ肷�邲�ƂɁA
$h\to 0$�̂Ƃ�$0$�Ɏ������Ă��܂��B�A�����̎�����A
$I_1$��$0$�Ɏ������Ă��邩�͂������ɂ�
������܂���B��ʂɋɌ��Ɛϕ��̏��������ɂ�
���ӂ���K�v������܂��B

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\begin{flushright}(�𓚗�쐬�F�O�p)
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