【微分積分】4-4-4 累乗根の積分｜問題集
1.次の不定積分を求めなさい。
(1)\(\displaystyle \int\frac{1}{1+\sqrt{x}}dx\)
\(t=\sqrt{x}\)とおくと、\(dx=2tdt\)
\(\displaystyle =\int\frac{2t}{1+t}dt\)
\(\displaystyle =\int\left(2-\frac{2}{1+t}\right)dt\)
\(\displaystyle =2t-2\log|1+t|+C\)
\(\displaystyle =2\sqrt{x}-2\log|1+\sqrt{x}|+C\)
\(\displaystyle =\int\frac{2t}{1+t}dt\)
\(\displaystyle =\int\left(2-\frac{2}{1+t}\right)dt\)
\(\displaystyle =2t-2\log|1+t|+C\)
\(\displaystyle =2\sqrt{x}-2\log|1+\sqrt{x}|+C\)
(2)\(\displaystyle \int\frac{\sqrt{x}}{\sqrt{x}+1}dx\)
\(t=\sqrt{x}\)とおくと、\(dx=2tdt\)
\(\displaystyle =\int\frac{t}{t+1}・2tdt\)
\(\displaystyle =\int\left(2t-2+\frac{2}{t+1}\right)dt\)
\(\displaystyle =t^2-2t+2\log|t+1|+C\)
\(\displaystyle =x-2\sqrt{x}+2\log|\sqrt{x}+1|+C\)
\(\displaystyle =\int\frac{t}{t+1}・2tdt\)
\(\displaystyle =\int\left(2t-2+\frac{2}{t+1}\right)dt\)
\(\displaystyle =t^2-2t+2\log|t+1|+C\)
\(\displaystyle =x-2\sqrt{x}+2\log|\sqrt{x}+1|+C\)
(3)\(\displaystyle \int\frac{1}{\sqrt{1-e^x}}dx\)
\(t=\sqrt{1-e^x}\)とおくと、\(\displaystyle dx=-\frac{2t}{e^x}dt\)
\(\displaystyle =\int\frac{1}{t}・\frac{2t}{t^2-1}dt\)
\(\displaystyle =\int\frac{2}{t^2-1}dt\)
\(\displaystyle =\log\left|\frac{t-1}{t+1}\right|+C\)
\(\displaystyle =2\log\left|\frac{\sqrt{1-e^x}-1}{\sqrt{1-e^x}+1}\right|+C\)
\(\displaystyle =\int\frac{1}{t}・\frac{2t}{t^2-1}dt\)
\(\displaystyle =\int\frac{2}{t^2-1}dt\)
\(\displaystyle =\log\left|\frac{t-1}{t+1}\right|+C\)
\(\displaystyle =2\log\left|\frac{\sqrt{1-e^x}-1}{\sqrt{1-e^x}+1}\right|+C\)
(4)\(\displaystyle \int\frac{x}{\sqrt{x-1}}dx\)
\(t=\sqrt{x-1}\)とおくと、\(dx=2tdt\)
\(\displaystyle =\int\frac{t^2+1}{t}・2tdt\)
\(\displaystyle =\int(2t^2+2)dt\)
\(\displaystyle =\frac{2t^3}{3}+2t+C\)
\(\displaystyle =\frac{2(x-1)^{\frac{3}{2}}}{3}+2\sqrt{x-1}+C\)
\(\displaystyle =\int\frac{t^2+1}{t}・2tdt\)
\(\displaystyle =\int(2t^2+2)dt\)
\(\displaystyle =\frac{2t^3}{3}+2t+C\)
\(\displaystyle =\frac{2(x-1)^{\frac{3}{2}}}{3}+2\sqrt{x-1}+C\)
(5)\(\displaystyle \int\frac{x}{\sqrt{x^2+4}}dx\)
\(t=\sqrt{x^2+4}\)とおくと、\(\displaystyle dx=\frac{t}{x}dt\)
\(\displaystyle =\int dt\)
\(\displaystyle =t+C\)
\(\displaystyle =\sqrt{x^2+4}+C\)
\(\displaystyle =\int dt\)
\(\displaystyle =t+C\)
\(\displaystyle =\sqrt{x^2+4}+C\)
(6)\(\displaystyle \int\frac{x^3}{\sqrt{x^2+4}}dx\)
\(t=\sqrt{x^2+4}\)とおくと、\(\displaystyle dx=\frac{t}{x}dt\)
\(\displaystyle =\int\frac{(t^2-4)t}{t}dt\)
\(\displaystyle =\int(t^2-4)dt\)
\(\displaystyle =\frac{t^3}{3}-4t+C\)
\(\displaystyle =\frac{(x^2+4)^{\frac{3}{2}}}{3}-4\sqrt{x^2+4}+C\)
\(\displaystyle =\int\frac{(t^2-4)t}{t}dt\)
\(\displaystyle =\int(t^2-4)dt\)
\(\displaystyle =\frac{t^3}{3}-4t+C\)
\(\displaystyle =\frac{(x^2+4)^{\frac{3}{2}}}{3}-4\sqrt{x^2+4}+C\)
(7)\(\displaystyle \int\frac{1}{\sqrt{x^2-2x-3}}dx\)
\(\displaystyle =\int\frac{1}{\sqrt{(x-1)^2-4}}dx\)
\(t=x-1\)とおくと、\(dx=dt\)
\(\displaystyle =\int\frac{1}{\sqrt{t^2-2^2}}dt\)
\(\displaystyle =\log|t+\sqrt{t^2-4}|+C\)
\(\displaystyle =\log|x-1+\sqrt{x^2-2x-3}|+C\)
\(t=x-1\)とおくと、\(dx=dt\)
\(\displaystyle =\int\frac{1}{\sqrt{t^2-2^2}}dt\)
\(\displaystyle =\log|t+\sqrt{t^2-4}|+C\)
\(\displaystyle =\log|x-1+\sqrt{x^2-2x-3}|+C\)
(8)\(\displaystyle \int\frac{\sqrt{x^2-1}}{x}dx\)
\(t=\sqrt{x^2-1}\)とおくと、\(\displaystyle dx=\frac{t}{x}dt\)
\(\displaystyle =\int\frac{t}{t^2+1}tdt\)
\(\displaystyle =\int\left(1-\frac{1}{t^2+1}\right)dt\)
\(\displaystyle =t-\tan^{-1}t+C\)
\(\displaystyle =\sqrt{x^2-1}-\tan^{-1}\sqrt{x^2-1}+C\)
\(\displaystyle =\int\frac{t}{t^2+1}tdt\)
\(\displaystyle =\int\left(1-\frac{1}{t^2+1}\right)dt\)
\(\displaystyle =t-\tan^{-1}t+C\)
\(\displaystyle =\sqrt{x^2-1}-\tan^{-1}\sqrt{x^2-1}+C\)
(9)\(\displaystyle \int x\sqrt{1+x}dx\)
\(t=\sqrt{1+x}\)とおくと、\(dx=2tdt\)
\(\displaystyle =\int(t^2-1)・t・2tdt\)
\(\displaystyle =\int(2t^4-2t^2)dt\)
\(\displaystyle =\frac{2t^5}{5}-\frac{2t^3}{3}+C\)
\(\displaystyle =\frac{2(1+x)^{\frac{5}{2}}}{5}-\frac{2(1+x)^{\frac{3}{2}}}{3}+C\)
\(\displaystyle =\int(t^2-1)・t・2tdt\)
\(\displaystyle =\int(2t^4-2t^2)dt\)
\(\displaystyle =\frac{2t^5}{5}-\frac{2t^3}{3}+C\)
\(\displaystyle =\frac{2(1+x)^{\frac{5}{2}}}{5}-\frac{2(1+x)^{\frac{3}{2}}}{3}+C\)
(10)\(\displaystyle \int\frac{\sqrt{x}}{\sqrt{x}-1}dx\)
\(t=\sqrt{x}\)とおくと、\(dx=2tdt\)
\(\displaystyle =\int\frac{t}{t-1}2tdt\)
\(\displaystyle =2\int\left(t+1+\frac{1}{t-1}\right)dt\)
\(\displaystyle =t^2+2t+2\log|t-1|+C\)
\(\displaystyle =x+2\sqrt{x}+2\log|\sqrt{x}-1|+C\)
\(\displaystyle =\int\frac{t}{t-1}2tdt\)
\(\displaystyle =2\int\left(t+1+\frac{1}{t-1}\right)dt\)
\(\displaystyle =t^2+2t+2\log|t-1|+C\)
\(\displaystyle =x+2\sqrt{x}+2\log|\sqrt{x}-1|+C\)
(11)\(\displaystyle \int\frac{1}{\sqrt{1+e^x}}dx\)
\(t=\sqrt{1+e^x}\)とおくと、\(\displaystyle dx=\frac{2t}{e^x}dt\)
\(\displaystyle =\int\frac{2t}{t(t^2-1)}dt\)
\(\displaystyle =2\int\frac{1}{t^2-1}dt\)
\(\displaystyle =\log\left|\frac{t-1}{t+1}\right|+C\)
\(\displaystyle =\log\left|\frac{\sqrt{1+e^x}-1}{\sqrt{1+e^x}+1}\right|+C\)
\(\displaystyle =\int\frac{2t}{t(t^2-1)}dt\)
\(\displaystyle =2\int\frac{1}{t^2-1}dt\)
\(\displaystyle =\log\left|\frac{t-1}{t+1}\right|+C\)
\(\displaystyle =\log\left|\frac{\sqrt{1+e^x}-1}{\sqrt{1+e^x}+1}\right|+C\)
(12)\(\displaystyle \int x^2\sqrt{x-1}dx\)
\(t=\sqrt{x-1}\)とおくと、\(dx=2tdt\)
\(\displaystyle =\int(t^2+1)^2・t・2tdt\)
\(\displaystyle =\int(2t^6+4t^4+2t^2)dt\)
\(\displaystyle =\frac{2t^7}{7}+\frac{4t^5}{5}+\frac{2t^3}{3}+C\)
\(\displaystyle =\frac{2(x-1)^{\frac{7}{2}}}{7}+\frac{4(x-1)^{\frac{5}{2}}}{5}+\frac{2(x-1)^{\frac{3}{2}}}{3}+C\)
\(\displaystyle =\int(t^2+1)^2・t・2tdt\)
\(\displaystyle =\int(2t^6+4t^4+2t^2)dt\)
\(\displaystyle =\frac{2t^7}{7}+\frac{4t^5}{5}+\frac{2t^3}{3}+C\)
\(\displaystyle =\frac{2(x-1)^{\frac{7}{2}}}{7}+\frac{4(x-1)^{\frac{5}{2}}}{5}+\frac{2(x-1)^{\frac{3}{2}}}{3}+C\)
(13)\(\displaystyle \int\sqrt{\frac{x+1}{x-1}}dx\)
\(\displaystyle t=\sqrt{\frac{x+1}{x-1}}\)とおくと、\(\displaystyle dx=-\frac{4t}{(t^2-1)^2}dt\)
\(\displaystyle =\int-\frac{4t^2}{(t^2-1)^2}dt\)
\(\displaystyle =\int\left(-\frac{1}{t+1}-\frac{1}{(t+1)^2}+\frac{1}{t-1}-\frac{1}{(t-1)^2}\right)dt\)
\(\displaystyle =\log\left|\frac{t-1}{t+1}\right|+\frac{2t}{t^2-1}+C\)
\(\displaystyle =\log\left|\frac{\sqrt{x+1}-\sqrt{x-1}}{\sqrt{x+1}+\sqrt{x-1}}\right|+\sqrt{x^2-1}+C\)
\(\displaystyle =\int-\frac{4t^2}{(t^2-1)^2}dt\)
\(\displaystyle =\int\left(-\frac{1}{t+1}-\frac{1}{(t+1)^2}+\frac{1}{t-1}-\frac{1}{(t-1)^2}\right)dt\)
\(\displaystyle =\log\left|\frac{t-1}{t+1}\right|+\frac{2t}{t^2-1}+C\)
\(\displaystyle =\log\left|\frac{\sqrt{x+1}-\sqrt{x-1}}{\sqrt{x+1}+\sqrt{x-1}}\right|+\sqrt{x^2-1}+C\)
(14)\(\displaystyle \int\frac{x}{\sqrt{x^2-4}}dx\)
\(t=x^2-4\)とおくと、\(\displaystyle dx=\frac{1}{2x}dt\)
\(\displaystyle =\int\frac{1}{2\sqrt{t}}dt\)
\(\displaystyle =\frac{1}{2}・2t^{\frac{1}{2}}+C\)
\(\displaystyle =\sqrt{x^2-4}+C\)
\(\displaystyle =\int\frac{1}{2\sqrt{t}}dt\)
\(\displaystyle =\frac{1}{2}・2t^{\frac{1}{2}}+C\)
\(\displaystyle =\sqrt{x^2-4}+C\)
(15)\(\displaystyle \int\frac{x^2}{\sqrt{4-x^2}}dx\)
\(x=2\sin t\)とおくと、\(\displaystyle dx=2\cos tdt\)
\(\displaystyle =\int\frac{4\sin^2t}{2\cos t}・2\cos tdt\)
\(\displaystyle =4\int\frac{1-\cos2t}{2}dt\)
\(\displaystyle =2t-2\sin t\cos t+C\)
\(\displaystyle =2\sin^{-1}\frac{x}{2}-\frac{x\sqrt{4-x^2}}{2}+C\)
\(\displaystyle =\int\frac{4\sin^2t}{2\cos t}・2\cos tdt\)
\(\displaystyle =4\int\frac{1-\cos2t}{2}dt\)
\(\displaystyle =2t-2\sin t\cos t+C\)
\(\displaystyle =2\sin^{-1}\frac{x}{2}-\frac{x\sqrt{4-x^2}}{2}+C\)
(16)\(\displaystyle \int\frac{e^x}{9-e^{2x}}dx\)
\(t=e^x\)とおくと、\(\displaystyle dx=\frac{1}{e^x}dt\)
\(\displaystyle =\int\frac{1}{9-t^2}dt\)
\(\displaystyle =\frac{1}{6}\int\left(\frac{1}{3-t}+\frac{1}{3+t}\right)dt\)
\(\displaystyle =\frac{1}{6}\log\left|\frac{3+t}{3-t}\right|+C\)
\(\displaystyle =\frac{1}{6}\log\left|\frac{3+e^x}{3-e^x}\right|+C\)
\(\displaystyle =\int\frac{1}{9-t^2}dt\)
\(\displaystyle =\frac{1}{6}\int\left(\frac{1}{3-t}+\frac{1}{3+t}\right)dt\)
\(\displaystyle =\frac{1}{6}\log\left|\frac{3+t}{3-t}\right|+C\)
\(\displaystyle =\frac{1}{6}\log\left|\frac{3+e^x}{3-e^x}\right|+C\)
(17)\(\displaystyle \int\frac{\sqrt{1-x^2}}{x^4}dx\)
\(x=\sin t\)とおくと、\(dx=\cos tdt\)
\(\displaystyle =\int\frac{\cos t}{\sin^4t}・\cos tdt\)
\(u=\tan t\)とおくと、
\(\displaystyle =\int\frac{\frac{1}{1+u^2}}{\frac{u^4}{(1+u^2)^2}}・\frac{1}{1+u^2}du\)
\(\displaystyle =\int\frac{1}{u^4}du\)
\(\displaystyle =-\frac{1}{3}u^{-3}+C\)
\(\displaystyle =-\frac{1}{3\tan^3t}+C\)
\(\displaystyle =-\frac{(1-x^2)^{\frac{3}{2}}}{3x^3}+C\)
\(\displaystyle =\int\frac{\cos t}{\sin^4t}・\cos tdt\)
\(u=\tan t\)とおくと、
\(\displaystyle =\int\frac{\frac{1}{1+u^2}}{\frac{u^4}{(1+u^2)^2}}・\frac{1}{1+u^2}du\)
\(\displaystyle =\int\frac{1}{u^4}du\)
\(\displaystyle =-\frac{1}{3}u^{-3}+C\)
\(\displaystyle =-\frac{1}{3\tan^3t}+C\)
\(\displaystyle =-\frac{(1-x^2)^{\frac{3}{2}}}{3x^3}+C\)
(18)\(\displaystyle \int\frac{1}{x^2\sqrt{x^2-a^2}}dx\)
\(x=a\sec t\)とおくと、\(dx=a\sec t\tan tdt\)
\(\displaystyle =\int\frac{a\sec t\tan t}{a^2\sec^2t・a\tan t}dt\)
\(\displaystyle =\frac{1}{a^2}\int\cos tdt\)
\(\displaystyle =\frac{1}{a^2}\sin t+C\)
\(\displaystyle =\frac{\sqrt{x^2-a^2}}{a^2x}+C\)
\(\displaystyle =\int\frac{a\sec t\tan t}{a^2\sec^2t・a\tan t}dt\)
\(\displaystyle =\frac{1}{a^2}\int\cos tdt\)
\(\displaystyle =\frac{1}{a^2}\sin t+C\)
\(\displaystyle =\frac{\sqrt{x^2-a^2}}{a^2x}+C\)
(19)\(\displaystyle \int\frac{1}{e^x\sqrt{4+e^{2x}}}dx\)
\(e^x=2\tan t\)とおくと、\(dx=\frac{2\sec^2t}{e^x}dt\)
\(\displaystyle =\int\frac{1}{2\tan t・2\sec t}・\frac{2\sec^2t}{2\sec t}dt\)
\(\displaystyle =\frac{1}{4}\int\frac{\cos t}{\sin^2t}dt\)
\(u=\sin t\)とおくと、\(du=\frac{1}{\cos t}dt\)
\(\displaystyle =\frac{1}{4}\int\frac{1}{u^2}du\)
\(\displaystyle =-\frac{1}{4u}+C\)
\(\displaystyle =-\frac{1}{4\sin t}+C\)
\(\displaystyle =-\frac{\sqrt{4+e^{2x}}}{4e^x}+C\)
\(\displaystyle =\int\frac{1}{2\tan t・2\sec t}・\frac{2\sec^2t}{2\sec t}dt\)
\(\displaystyle =\frac{1}{4}\int\frac{\cos t}{\sin^2t}dt\)
\(u=\sin t\)とおくと、\(du=\frac{1}{\cos t}dt\)
\(\displaystyle =\frac{1}{4}\int\frac{1}{u^2}du\)
\(\displaystyle =-\frac{1}{4u}+C\)
\(\displaystyle =-\frac{1}{4\sin t}+C\)
\(\displaystyle =-\frac{\sqrt{4+e^{2x}}}{4e^x}+C\)
(20)\(\displaystyle \int\frac{x}{\sqrt{6x-x^2}}dx\)
\(\displaystyle =\int\frac{x}{\sqrt{3^2-(3-x)^2}}dx\)
\(3-x=3\sin t\)とおくと、\(dx=-3\cos tdt\)
\(\displaystyle =\int\frac{3-3\sin t}{3\cos t}・-3\cos tdt\)
\(\displaystyle =-3\int(1-\sin t)dt\)
\(\displaystyle =-3t-3\cos t+C\)
\(\displaystyle =-3\sin^{-1}\frac{3-x}{3}-3\frac{\sqrt{9-(3-x)^2}}{3}+C\)
\(\displaystyle =-3\sin^{-1}\frac{3-x}{3}-\sqrt{6x-x^2}+C\)
\(3-x=3\sin t\)とおくと、\(dx=-3\cos tdt\)
\(\displaystyle =\int\frac{3-3\sin t}{3\cos t}・-3\cos tdt\)
\(\displaystyle =-3\int(1-\sin t)dt\)
\(\displaystyle =-3t-3\cos t+C\)
\(\displaystyle =-3\sin^{-1}\frac{3-x}{3}-3\frac{\sqrt{9-(3-x)^2}}{3}+C\)
\(\displaystyle =-3\sin^{-1}\frac{3-x}{3}-\sqrt{6x-x^2}+C\)
(21)\(\displaystyle \int\frac{x}{\sqrt{x^2-2x-3}}dx\)
\(\displaystyle =\int\frac{x}{\sqrt{(x-1)^2-2^2}}dx\)
\(x-1=2\sec t\)とおくと、\(dx=2\sec t\tan tdt\)
\(\displaystyle =\int\frac{2\sec t+1}{2\tan t}・2\sec t\tan tdt\)
\(\displaystyle =\int(2\sec^2t+\sec t)dt\)
\(\displaystyle =2\tan t+\log|\sec t+\tan t|+C\)
\(\displaystyle =\sqrt{(x-1)^2-4}\)
\(\displaystyle \ \ \ +\log\left|\frac{x-1}{2}+\frac{\sqrt{(x-1)^2-4}}{2}\right|+C\)
\(\displaystyle =\sqrt{x^2-2x-3}\)
\(\displaystyle \ \ \ +\log\left|\frac{x-1}{2}+\frac{\sqrt{(x-1)^2-4}}{2}\right|+C\)
\(x-1=2\sec t\)とおくと、\(dx=2\sec t\tan tdt\)
\(\displaystyle =\int\frac{2\sec t+1}{2\tan t}・2\sec t\tan tdt\)
\(\displaystyle =\int(2\sec^2t+\sec t)dt\)
\(\displaystyle =2\tan t+\log|\sec t+\tan t|+C\)
\(\displaystyle =\sqrt{(x-1)^2-4}\)
\(\displaystyle \ \ \ +\log\left|\frac{x-1}{2}+\frac{\sqrt{(x-1)^2-4}}{2}\right|+C\)
\(\displaystyle =\sqrt{x^2-2x-3}\)
\(\displaystyle \ \ \ +\log\left|\frac{x-1}{2}+\frac{\sqrt{(x-1)^2-4}}{2}\right|+C\)
(22)\(\displaystyle \int\sqrt{6x-x^2-8}dx\)
\(\displaystyle =\int\sqrt{1-(3-x)^2}dx\)
\(3-x=\sin t\)とおくと、\(dx=-\cos tdt\)
\(\displaystyle =-\int\cos^2tdt\)
\(\displaystyle =-\int\frac{1+\cos t}{2}dt\)
\(\displaystyle =-\frac{t}{2}-\frac{\sin2t}{4}+C\)
\(\displaystyle =-\frac{t}{2}-\frac{\sin t\cos t}{4}+C\)
\(\displaystyle =-\frac{\sin^{-1}(3-x)}{2}-\frac{(3-x)\sqrt{1-(3-x)^2}}{2}+C\)
\(3-x=\sin t\)とおくと、\(dx=-\cos tdt\)
\(\displaystyle =-\int\cos^2tdt\)
\(\displaystyle =-\int\frac{1+\cos t}{2}dt\)
\(\displaystyle =-\frac{t}{2}-\frac{\sin2t}{4}+C\)
\(\displaystyle =-\frac{t}{2}-\frac{\sin t\cos t}{4}+C\)
\(\displaystyle =-\frac{\sin^{-1}(3-x)}{2}-\frac{(3-x)\sqrt{1-(3-x)^2}}{2}+C\)
(23)\(\displaystyle \int x\sqrt{x^2+6x}dx\)
\(\displaystyle =\int\sqrt{(x+3)^2-9}dx\)
\(t=x+3\)とおくと、\(dx=dt\)
\(\displaystyle =\int(t-3)\sqrt{t^2-9}dt\)
\(\displaystyle =\frac{(t^2-9)^{\frac{3}{2}}}{3}\)
\(\displaystyle \ \ \ -3\left(\frac{t}{2}\sqrt{t^2-9}-\frac{9}{2}\log\left|t+\sqrt{t^2-9}\right|\right)\)
\(\displaystyle =\frac{(x^2+6x)^{\frac{3}{2}}}{3}-\frac{3}{2}(x+3)\sqrt{x^2+6x}\)
\(\displaystyle \ \ \ +\frac{27}{2}\log\left|x+3+\sqrt{x^2+6x}\right|+C\)
\(t=x+3\)とおくと、\(dx=dt\)
\(\displaystyle =\int(t-3)\sqrt{t^2-9}dt\)
\(\displaystyle =\frac{(t^2-9)^{\frac{3}{2}}}{3}\)
\(\displaystyle \ \ \ -3\left(\frac{t}{2}\sqrt{t^2-9}-\frac{9}{2}\log\left|t+\sqrt{t^2-9}\right|\right)\)
\(\displaystyle =\frac{(x^2+6x)^{\frac{3}{2}}}{3}-\frac{3}{2}(x+3)\sqrt{x^2+6x}\)
\(\displaystyle \ \ \ +\frac{27}{2}\log\left|x+3+\sqrt{x^2+6x}\right|+C\)
次の学習に進もう！