0.0.0.1 問題.

以下の広義積分を計算せよ。 (1) $\dsp\int_{1}^\infty\frac{\Dx}{x^2}$     (2) $\dsp\int_{0}^1\frac{\Dx}{\sqrt{x}}$     (3) $\dsp\int_{1}^\infty\frac{\Dx}{x}$     (4) $\dsp\int_{1}^\infty\frac{\Dx}{1+x^2}$    

(1)

$$\int_{1}^\infty\frac{\Dx}{x^2} =\lim_{R\to\infty}\int_1^R x^{-2}\;\Dx =\lim_{R\to\infty}\left[- x^{-1}\right]_1^R =\lim_{R\to\infty}(1-R^{-1})=1.$$

(2)

$$\dsp\int_{0}^1\frac{\Dx}{\sqrt{x}} =\lim_{\eps\to+0}\int_\eps^1 x... ..._{\eps\to+0}\left[2x^{1/2}\right]_\eps^1 =\lim_{\eps\to+0} 2(1-\eps^{1/2}) =2.$$

(3)

$$\int_{1}^\infty\frac{\Dx}{x} =\lim_{R\to\infty}\int_1^R \frac{1}{... ...o\infty}\left[\log\vert x\vert\right]_1^R =\lim_{R\to\infty}(\log R-0)=\infty.$$

(4)

$$\int_{1}^\infty\frac{\Dx}{1+x^2} =\lim_{R\to\infty}\int_1^R \frac... ...R\to\infty}(\tan^{-1}R-\tan^{-1}1) =\frac{\pi}{2}-\frac{\pi}{4}=\frac{\pi}{4}.$$