1.次の不定積分を求めなさい。
(1)\(\displaystyle \int x(1-x)^4dx\)
\(1-x=t\)とおくと、\(\displaystyle dx=-dt\)なので、
\(\displaystyle =\int(t-1)t^4dt\)
\(\displaystyle =\int(t^5-t^4)dt\)
\(\displaystyle =\frac{1}{6}t^6-\frac{1}{5}t^5+C\)
\(\displaystyle =-\frac{1}{30}(1-x)^5(5x+1)+C\ \ \)(\(C\)は積分定数)
(2)\(\displaystyle \int x\sqrt{2x-1}dx\)
\(\sqrt{2x-1}=t\)とおくと、\(\displaystyle dx=tdt\)なので、
\(\displaystyle =\int\frac{1}{2}(t^2+1)t・tdt\)
\(\displaystyle =\frac{1}{2}\int(t^4+t^2)dt\)
\(\displaystyle =\frac{1}{2}\left(\frac{1}{5}t^5+\frac{1}{3}t^3\right)+C\)
\(\displaystyle =\frac{1}{15}(3x+1)(2x-1)\sqrt{2x-1}+C\ \ \)(\(C\)は積分定数)
(3)\(\displaystyle \int\frac{x}{\sqrt{x+1}}dx\)
\(\sqrt{x+1}=t\)とおくと、\(\displaystyle dx=2tdt\)なので、
\(\displaystyle =\int\frac{t^2-1}{t}・2tdt\)
\(\displaystyle =2\int(t^2-1)dt\)
\(\displaystyle =\frac{2}{3}t^3-2t+C\)
\(\displaystyle =\frac{2}{3}(x-2)\sqrt{x+1}+C\ \ \)(\(C\)は積分定数)
(4)\(\displaystyle \int\sqrt[4]{2x-3}dx\)
\(2x-3=t\)とおくと、\(\displaystyle dx=\frac{1}{2}dt\)なので、
\(\displaystyle =\int\sqrt[4]{t}・\frac{1}{2}dt\)
\(\displaystyle =\frac{1}{2}\int t^{\frac{1}{4}}dt\)
\(\displaystyle =\frac{2}{5}t^{\frac{5}{4}}+C\)
\(\displaystyle =\frac{2}{5}(2x-3)\sqrt[4]{2x-3}+C\ \ \)(\(C\)は積分定数)
(5)\(\displaystyle \int\cos(5x+1)dx\)
\(5x+1=t\)とおくと、\(\displaystyle dx=\frac{1}{5}dt\)なので、
\(\displaystyle =\int\cos t・\frac{1}{5}dt\)
\(\displaystyle =\frac{1}{5}\int\cos tdt\)
\(\displaystyle =\frac{1}{5}\sin t+C\)
\(\displaystyle =\frac{1}{5}\sin(5x+1)+C\ \ \)(\(C\)は積分定数)
(6)\(\displaystyle \int\frac{x^2}{(x-2)^2}dx\)
\(x-2=t\)とおくと、\(dx=dt\)なので、
\(\displaystyle =\int\frac{(t+2)^2}{t^2}dt\)
\(\displaystyle =\int\left(1+\frac{4}{t}+\frac{4}{t^2}\right)dt\)
\(\displaystyle =t+4\log|t|-\frac{4}{t}+C\)
\(\displaystyle =x+4\log|x-2|-\frac{4}{x-2}+C\ \ \)(\(C\)は積分定数)
(7)\(\displaystyle \int x\sqrt{x+2}dx\)
\(x+2=t\)とおくと、\(dx=dt\)なので、
\(\displaystyle =\int(t-2)\sqrt{t}dt\)
\(\displaystyle =\int(t^{\frac{3}{2}}-2t^{\frac{1}{2}})dt\)
\(\displaystyle =\frac{2}{5}t^{\frac{5}{2}}-\frac{4}{3}t^{\frac{3}{2}}+C\)
\(\displaystyle =\frac{2}{15}(x+2)(3x-4)\sqrt{x+2}+C\ \ \)(\(C\)は積分定数)
(8)\(\displaystyle \int x^2\sqrt{x^3+2}dx\)
\(x^3+2=t\)とおくと、\(\displaystyle dx=\frac{1}{3x^2}dt\)なので、
\(\displaystyle =\frac{1}{3}\int\sqrt{t}dt\)
\(\displaystyle =\frac{1}{3}・\frac{2}{3}t^{\frac{3}{2}}+C\)
\(\displaystyle =\frac{2}{9}(x^3+2)\sqrt{x^3+2}+C\ \ \)(\(C\)は積分定数)
(9)\(\displaystyle \int\sin^3x\cos xdx\)
\(\sin x=t\)とおくと、\(\displaystyle dx=\frac{1}{\cos x}dt\)なので、
\(\displaystyle =\int t^3dt\)
\(\displaystyle =\frac{1}{4}t^4+C\)
\(\displaystyle =\frac{1}{4}\sin^4x+C\ \ \)(\(C\)は積分定数)
(10)\(\displaystyle \int\frac{\log x}{x}dx\)
\(\log x=t\)とおくと、\(\displaystyle dx=xdt\)なので、
\(\displaystyle =\int tdt\)
\(\displaystyle =\frac{1}{2}t^2+C\)
\(\displaystyle =\frac{1}{2}(\log x)^2+C\ \ \)(\(C\)は積分定数)
(11)\(\displaystyle \int\frac{e^x}{(e^x-3)^2}dx\)
\(\log e^x-3=t\)とおくと、\(\displaystyle e^xdx=dt\)なので、
\(\displaystyle =\int\frac{1}{t^2}dt\)
\(\displaystyle =-\frac{1}{t}+C\)
\(\displaystyle =-\frac{1}{e^x-3}+C\ \ \)(\(C\)は積分定数)
(12)\(\displaystyle \int\cos^2x\sin xdx\)
\(\cos x=t\)とおくと、\(\displaystyle -\sin xdx=dt\)なので、
\(\displaystyle =-\int t^2dt\)
\(\displaystyle =-\frac{1}{3}t^3+C\)
\(\displaystyle =-\frac{1}{3}\cos^3x+C\ \ \)(\(C\)は積分定数)
(13)\(\displaystyle \int\frac{3x^2}{\sqrt[3]{x^3+2}}dx\)
\(x^3+2=t\)とおくと、\(\displaystyle 3x^2dx=dt\)なので、
\(\displaystyle =\int t^{-\frac{1}{3}}dt\)
\(\displaystyle =\frac{3}{2}\sqrt[3]{t^2}+C\)
\(\displaystyle =\frac{3}{2}\sqrt[3]{(x^3+2)^2}+C\ \ \)(\(C\)は積分定数)
(14)\(\displaystyle \int\frac{2x+1}{x^2+x-1}dx\)
\(x^2+x-1=t\)とおくと、\(\displaystyle (2x+1)dx=dt\)なので、
\(\displaystyle =\int\frac{1}{t}dt\)
\(\displaystyle =\log|t|+C\)
\(\displaystyle =\log|x^2+x-1|+C\ \ \)(\(C\)は積分定数)
(15)\(\displaystyle \int\frac{e^x}{e^x+1}dx\)
\(e^x+1=t\)とおくと、\(\displaystyle e^xdx=dt\)なので、
\(\displaystyle =\int\frac{1}{t}dt\)
\(\displaystyle =\log|t|+C\)
\(\displaystyle =\log|e^x+1|+C\ \ \)(\(C\)は積分定数)
(16)\(\displaystyle \int\frac{dx}{\tan x}\)
\(\displaystyle =\int\frac{\cos x}{\sin x}dx\)
\(\sin x=t\)とおくと、\(\displaystyle \cos xdx=dt\)なので、
\(\displaystyle =\int\frac{1}{t}dt\)
\(\displaystyle =\log|t|+C\)
\(\displaystyle =\log|\sin x|+C\ \ \)(\(C\)は積分定数)
(17)\(\displaystyle \int\frac{3x^2}{x^3+2}dx\)
\(x^3+2=t\)とおくと、\(\displaystyle 3x^2dx=dt\)なので、
\(\displaystyle =\int\frac{1}{t}dt\)
\(\displaystyle =\log|t|+C\)
\(\displaystyle =\log|x^3+2|+C\ \ \)(\(C\)は積分定数)
(18)\(\displaystyle \int x\sin xdx\)
\(\displaystyle =-x\cos x-\int(x)'(-\cos x)dx\)
\(\displaystyle =-x\cos x+\int\cos xdx\)
\(\displaystyle =-x\cos x+\sin x+C\ \ \)(\(C\)は積分定数)
(19)\(\displaystyle \int xe^{-x}dx\)
\(\displaystyle =-xe^{-x}-\int(x)'(-e^{-x})dx\)
\(\displaystyle =-xe^{-x}+\int e^{-x}dx\)
\(\displaystyle =-(x+1)e^{-x}+C\ \ \)(\(C\)は積分定数)
(20)\(\displaystyle \int x\cos xdx\)
\(\displaystyle =x\sin x-\int(x)'\sin xdx\)
\(\displaystyle =x\sin x+\int\sin xdx\)
\(\displaystyle =x\sin x+\cos x+C\ \ \)(\(C\)は積分定数)
(21)\(\displaystyle \int\log2xdx\)
\(\displaystyle =x\log2x-\int x(\log2x)'dx\)
\(\displaystyle =x\log2x-\int dx\)
\(\displaystyle =x\log2x-x+C\ \ \)(\(C\)は積分定数)
(22)\(\displaystyle \int\log x^2dx\)
\(\displaystyle =x\log x^2-\int x(\log x^2)'dx\)
\(\displaystyle =x\log x^2-2\int dx\)
\(\displaystyle =x\log x^2-2x+C\ \ \)(\(C\)は積分定数)
(23)\(\displaystyle \int x\log xdx\)
\(\displaystyle =\frac{x^2}{2}\log x-\frac{1}{2}\int x^2(\log x)'dx\)
\(\displaystyle =\frac{x^2}{2}\log x-\frac{1}{2}\int xdx\)
\(\displaystyle =\frac{x^2}{2}\log x-\frac{1}{4}x^2+C\ \ \)(\(C\)は積分定数)
(24)\(\displaystyle \int x^2\log xdx\)
\(\displaystyle =\frac{x^3}{3}\log x-\frac{1}{3}\int x^3(\log x)'dx\)
\(\displaystyle =\frac{x^3}{3}\log x-\frac{1}{3}\int x^2dx\)
\(\displaystyle =\frac{x^3}{3}\log x-\frac{1}{9}x^3+C\ \ \)(\(C\)は積分定数)
(25)\(\displaystyle \int\frac{x^2}{e^x}dx\)
\(\displaystyle =-x^2e^{-x}-\int(x^2)'(-e^{-x})dx\)
\(\displaystyle =-\frac{x^2}{e^x}+2\int xe^{-x}dx\)
\(\displaystyle =-\frac{x^2}{e^x}+2\{-xe^{-x}-\int(x)'(-e^{-x})dx\}\)
\(\displaystyle =-\frac{x^2}{e^x}-2\{xe^{-x}+\int e^{-x}dx\}\)
\(\displaystyle =-\frac{x^2+2x+2}{e^x}+C\ \ \)(\(C\)は積分定数)
