6.32 演習～広義積分

問 6.171 (広義積分) 次の広義積分の値を求めよ.
    (1)   $\displaystyle{\int_{0}^{1}\!\!\frac{dx}{\sqrt{x}}}$     (2)   $\displaystyle{\int_{0}^{1}\frac{dx}{x}}$     (3)   $\displaystyle{\int_{0}^{1}\frac{dx}{x^2}}$     (4)   $\displaystyle{\int_{0}^{1}\frac{dx}{x^p}}$ ()     (5)   $\displaystyle{\int_{0}^{1}\frac{dx}{x^p}}$ ( $p\ge1$ )
    (6)   $\displaystyle{\int_{1}^{\infty}\!\!\frac{dx}{\sqrt{x}}}$     (7)   $\displaystyle{\int_{1}^{\infty}\frac{dx}{x}}$     (8)   $\displaystyle{\int_{1}^{\infty}\frac{dx}{x^2}}$     (9)   $\displaystyle{\int_{1}^{\infty}\frac{dx}{x^p}}$ ( $0<p\le1$ )     (10)   $\displaystyle{\int_{1}^{\infty}\frac{dx}{x^p}}$ ()
    (11)   $\displaystyle{\int_{-\infty}^{0}e^{\alpha x}\,\,dx}$ ( $\alpha>0$ )     (12)   $\displaystyle{\int_{-\infty}^{0}e^{\alpha x}\,\,dx}$ ( $\alpha<0$ )     (13)   $\displaystyle{\int_{0}^{\infty}e^{\alpha x}\,\,dx}$ ( $\alpha>0$ )
    (14)   $\displaystyle{\int_{0}^{\infty}e^{\alpha x}\,\,dx}$ ( $\alpha<0$ )     (15)   $\displaystyle{\int_{0}^{\infty}\!\frac{dx}{1+x^2}}$     (16)   $\displaystyle{\int_{0}^{1}x\log x\,\,dx}$     (17)   $\displaystyle{\int_{0}^{1}x^2\log x\,\,dx}$
    (18)   $\displaystyle{\int_{1}^{2}\frac{dx}{\sqrt{2-x}}}$     (19)   $\displaystyle{\int_{0}^{3}\frac{dx}{\sqrt{3-x}}}$     (20)   $\displaystyle{\int_{-\infty}^{-1}\frac{dx}{x^2}}$     (21)   $\displaystyle{\int_{-1}^{2}\frac{dx}{\sqrt{\vert x\vert}}}$     (22)   $\displaystyle{\int_{-1}^{1}\frac{dx}{\sqrt{1-x^2}}}$
    (23)   $\displaystyle{\int_{0}^{r}\frac{4r}{\sqrt{r^2-x^2}}dx}$ ()     (24)   $\displaystyle{\int_{1}^{2}\frac{dx}{\sqrt{x^2-1}}}$     (25)   $\displaystyle{\int_{1}^{\infty}\frac{dx}{x\sqrt{x^2-1}}}$     (26)   $\displaystyle{\int_{0}^{2}\frac{dx}{\sqrt{\vert x^2-1\vert}}}$
    (27)   $\displaystyle{\int_{0}^{2}\frac{2x-3}{\sqrt{\vert x^2-3x+2\vert}}dx}$     (28)   $\displaystyle{\int_{0}^{\pi/2}\frac{\cos x}{\sqrt{\sin x}}dx}$     (29)   $\displaystyle{\int_{0}^{\infty}xe^{-x}}$     (30)   $\displaystyle{\int_{0}^{\infty}xe^{-x^2}}$
    (31)   $\displaystyle{\int_{0}^{\infty}\frac{dx}{e^{x}+e^{-x}}}$

問 6.172 (広義積分) 定積分 $\displaystyle{I(a,b)=\int_{a}^{b}x\,dx}$ を計算し，条件 () を課した後に極限 $\displaystyle{\lim_{b\to+\infty}I(-kb,b)}$ を求める．が次の範囲内にあるとき極限をそれぞれ求めよ．
(1) (2) (3)

問 6.173 (コーシーの主値積分) 次の広義積分とコーシー主値積分の違いに注意して積分の値を求めよ.
(1) $\displaystyle{\int_{-\infty}^{\infty}x\,dx}$ (2) $\displaystyle{\text{v.p.}\int_{-\infty}^{\infty}x\,dx}$ (3) $\displaystyle{\int_{-1}^{1}\frac{dx}{x}}$ (4) $\displaystyle{\text{v.p.}\int_{-1}^{1}\frac{dx}{x}}$ (5) $\displaystyle{\int_{-1}^{1}\frac{dx}{x^2}}$
(6) $\displaystyle{\text{v.p.}\int_{-1}^{1}\frac{dx}{x^2}}$ (7) $\displaystyle{\int_{-1}^{1}\frac{dx}{\sqrt{\vert x\vert}}}$ (8) $\displaystyle{\text{v.p.}\int_{-1}^{1}\frac{dx}{\sqrt{\vert x\vert}}}$ (9) $\displaystyle{\int_{-\infty}^{\infty}\!\frac{dx}{1+x^2}}$ (10) $\displaystyle{\text{v.p.}\int_{-\infty}^{\infty}\!\frac{dx}{1+x^2}}$

問 6.174 (広義積分) 広義積分を用いて次の級数が収束するか発散するか議論せよ．
(1) $\displaystyle{\sum_{n=1}^{\infty}\frac{1}{n}}$ (2) $\displaystyle{\sum_{n=1}^{\infty}\frac{1}{\sqrt{n}}}$ (3) $\displaystyle{\sum_{n=1}^{\infty}\frac{1}{n^2}}$ (4) $\displaystyle{\sum_{n=1}^{\infty}\frac{1}{n^p}}$ ( $0<p\le1$ ) (5) $\displaystyle{\sum_{n=1}^{\infty}\frac{1}{n^p}}$ ()

問 6.175 (広義積分) 次の広義積分は収束するか発散するか議論せよ．
(1) $\displaystyle{\int_{0}^{1}\frac{\sin x}{\sqrt{1-x}}\,dx}$ (2) $\displaystyle{\int_{0}^{1}\sin\frac{1}{x}\,dx}$ (3) $\displaystyle{\int_{1}^{\infty}\frac{dx}{x\sqrt{x-1}}}$ (4) $\displaystyle{\int_{2}^{\infty}\frac{dx}{\sqrt[3]{x(x-1)}}}$ (5) $\displaystyle{\int_{0}^{1}\log x\,dx}$
(6) $\displaystyle{\int_{0}^{\infty}e^{-x^2}\,dx}$ (7) $\displaystyle{\int_{0}^{\pi/2}\frac{dx}{\sin x}}$ (8) $\displaystyle{\int_{0}^{1}\frac{dx}{\sqrt{x(1-x)}}}$ (9) $\displaystyle{\int_{-\infty}^{\infty}\frac{dx}{\sqrt{x^4+1}}}$ (10) $\displaystyle{\int_{0}^{\infty}\frac{dx}{\sqrt{x^2+1}}}$

平成22年6月17日

6.32 演習 ～ 広義積分

6.32 演習～広義積分