次の不定積分を求めよ．
(1) \(\displaystyle\int \frac{(\sqrt{x}+1)^{2}}{x}dx\)
(2) \(\displaystyle\int\sqrt{x}(x-1)^{2}dx\)
(3) \(\displaystyle\int\frac{(x-2)^{2}}{x^{4}}\)
(4) \(\displaystyle\int \left(\frac{x-1}{x}\right)^{2}dx\)
(5) \(\displaystyle\int\frac{(\sqrt{x}+2)^{2}}{x^{2}}dx\)
(6) \(\displaystyle\int\left( x+\frac{1}{x}\right)^{3}dx\)
分子の式を展開した後，分母と割り算をすることで積分しやすくしましょう（分子の次数を下げます）
式によっては，展開する前に分子で割り算できれば，最初に割り算をしているものもあります。
【解答】\(C\) は積分定数
(1) \(\displaystyle\int \frac{(\sqrt{x}+1)^{2}}{x}dx\)
\(\displaystyle＝\int \frac{(x+2\sqrt{x}+1)}{x}dx\)
\(\displaystyle＝\int (1+2x^{-\frac{1}{2}}+x^{-1})dx\)
\(\displaystyle＝x+4x^{\frac{1}{2}}+\log|x|+C\)
(2) \(\displaystyle\int\sqrt{x}(x-1)^{2}dx\)
\(\displaystyle＝\int \frac{(\sqrt{x}+1)^{2}}{x}dx\)
\(\displaystyle＝\int (x^{\frac{5}{2}}-2x^{\frac{3}{2}}+x^{\frac{1}{2}})dx\)
\(\displaystyle＝\frac{2}{7}x^{\frac{7}{2}}-\frac{4}{5}2x^{\frac{5}{2}}+\frac{2}{3}x^{\frac{3}{2}}+C\)
(3) \(\displaystyle\int\frac{(x-2)^{2}}{x^{4}}\)
\(\displaystyle＝\int\frac{x^{2}-4x+4}{x^{4}}dx\)
\(\displaystyle＝\int x^{-2}-4x^{-3}+4x^{-4}dx\)
\(\displaystyle＝-x^{-1}-4・(-\frac{1}{2})x^{-2}+(-\frac{4}{3})x^{-3}+C\)
\(\displaystyle＝-x^{-1}+2x^{-2}-\frac{4}{3}x^{-3}+C\)
(4) \(\displaystyle\int \left(\frac{x-1}{x}\right)^{2}dx\)
\(\displaystyle＝\int(1-\frac{1}{x})^{2}dx\)
\(\displaystyle＝\int(1-\frac{2}{x}+x^{-2})dx\)
\(\displaystyle＝x-2\log|x|-x^{-1}+C\)
(5) \(\displaystyle\int\frac{(\sqrt{x}+2)^{2}}{x^{2}}dx\)
\(\displaystyle＝\int\frac{(x\sqrt{x}+6x+12\sqrt{x}+8}{x^{2}}dx\)
\(\displaystyle＝\int（x^{-\frac{1}{2}}+6x^{-1}+12x^{\frac{-3}{2}}+8x^{-2})dx\)
\(\displaystyle＝2x^{\frac{1}{2}}+6\log|x|+12(-2)x^{-\frac{1}{2}}+8(-1)x^{-1}+C\)
\(\displaystyle＝2x^{\frac{1}{2}}+6\log|x|-24x^{-\frac{1}{2}}-8x^{-1}+C\)
(6) \(\displaystyle\int\left( x+\frac{1}{x}\right)^{3}dx\)
\(\displaystyle＝\int(x^{3}+3x+3x^{-1}+x^{-3})dx\)
\(\displaystyle＝\frac{1}{4}x^{4}+\frac{3}{2}x^{2}+3\log|x|-\frac{1}{2}x^{-2}+C\)
