【微分積分】4-3-1 積分公式の利用｜問題集

\(t=2-x\)とおくと、\(\displaystyle dx=-dt\)
\(\displaystyle =-\int e^tdt\)
\(\displaystyle =-e^{2-x}+C\)

\(t=1-x\)とおくと、\(\displaystyle dx=-dt\)
\(\displaystyle =-\int\sec^2tdt\)
\(\displaystyle =-\tan(1-x)+C\)

\(t=1-x^2\)とおくと、\(\displaystyle dx=-\frac{1}{2x}dt\)
\(\displaystyle =-\frac{1}{2}\int\frac{1}{\sqrt{t}}dt\)
\(\displaystyle =-\frac{1}{2}\int t^{-\frac{1}{2}}dt\)
\(\displaystyle =-\sqrt{1-x^2}+C\)

\(t=\cos x\)とおくと、\(\displaystyle dx=-\frac{1}{\sin x}dt\)
\(\displaystyle =-\int\frac{1}{t^2}dt\)
\(\displaystyle =-\int t^{-2}dt\)
\(\displaystyle =\frac{1}{t}+C\)
\(\displaystyle =\sec x+C\)

\(\displaystyle t=\frac{1}{x}\)とおくと、\(dx=-x^2dt\)
\(\displaystyle =-\int e^tdt\)
\(\displaystyle =-e^t+C\)
\(\displaystyle =-e^{\frac{1}{x}}+C\)

\(t=3\tan\theta+1\)とおくと、\(\displaystyle d\theta=\frac{1}{3\sec^2\theta}dt\)
\(\displaystyle =\frac{1}{3}\int\frac{1}{\sqrt{t}}dt\)
\(\displaystyle =\frac{2}{3}\sqrt{t}+C\)
\(\displaystyle =\frac{2\sqrt{1+3\tan\theta}}{3}+C\)

\(\displaystyle =\int\frac{\cos^2x}{\sin^2x}dx\)
\(\displaystyle =2\int\frac{1-\sin^2x}{\sin^2x}dx\)
\(\displaystyle =2\int(\csc^2x-1)dx\)
\(\displaystyle =-2(\cot x+x)+C\)

\(t=\log x\)とおくと、\(dx=xdt\)
\(\displaystyle =\int tdt\)
\(\displaystyle =\frac{1}{t^2}+C\)
\(\displaystyle =\frac{1}{2}(\log x)^2+C\)

\(t=e^x\)とおくと、\(\displaystyle dx=\frac{1}{e^x}dt\)
\(\displaystyle =\int\frac{1}{1+t^2}dt\)
\(\displaystyle =\tan^{-1}e^x+C\)

\(t=x^2\)とおくと、\(\displaystyle dx=\frac{1}{2x}dt\)
\(\displaystyle =\frac{1}{2}\int\sin tdt\)
\(\displaystyle =-\frac{\cos x^2}{2}+C\)

\(\displaystyle =x\sin^{-1}x-\int x・\frac{1}{\sqrt{1-x^2}}dx\)
\(t=1-x^2\)とおくと、\(\displaystyle dx=-\frac{1}{2x}dt\)
\(\displaystyle =x\sin^{-1}x-\frac{1}{2}\int t^{-\frac{1}{2}}dt\)
\(\displaystyle =x\sin^{-1}x-t^{\frac{1}{2}}+C\)
\(\displaystyle =x\sin^{-1}x+\sqrt{1-x^2}+C\)

\(\displaystyle =\frac{x^2}{2}\log x-\int\frac{x^2}{2}・\frac{1}{x}dx\)
\(\displaystyle =\frac{x^2\log x}{2}-\int\frac{x}{2}dx\)
\(\displaystyle =\frac{x^2\log x}{2}-\frac{x^2}{4}+C\)

\(\displaystyle =-x^2e^{-x}+2\int xe^{-x}dx\)
\(\displaystyle =-x^2e^{-x}+2\left(-xe^{-x}+\int e^{-x}dx\right)\)
\(\displaystyle =-x^2e^{-x}+2(-xe^{-x}-e^{-x}+C)\)
\(\displaystyle =-e^x(x^2+2x+2)+C\)

\(\displaystyle =x(\log x)^2-2\int\log xdx\)
\(\displaystyle =x(\log x)^2-2\left(x\log x-\int x・\frac{1}{x}dx\right)\)
\(\displaystyle =x(\log x)^2-2(x\log x-x+C)\)
\(\displaystyle =x(\log x)^2-2(x\log x-x)+C\)

\(t=x+5\)とおくと、\(\displaystyle dx=dt\)
\(\displaystyle =\int(t-5)t^{14}dt\)
\(\displaystyle =\int(t^{15}-5t^{14})dt\)
\(\displaystyle =\frac{1}{16}t^{16}-\frac{1}{3}t^{15}+C\)
\(\displaystyle =\frac{1}{16}(x+5)^{16}-\frac{1}{3}(x+5)^{15}+C\)

\(\displaystyle =x^2\sin x-\int2x\sin xdx\)
\(\displaystyle =x^2\sin x-(-2x\cos x+\int2\cos xdx)\)
\(\displaystyle =x^2\sin x-2(-x\cos x+\sin x+C)\)
\(\displaystyle =x^2\sin x+2(x\cos x-\sin x)+C\)

\(\displaystyle =e^x\sin x-\int e^x\cos xdx\)
\(\displaystyle =e^x\sin x-(e^x\cos x-\int-e^x\sin xdx)\)
\(\displaystyle =e^x\sin x-e^x\cos x-\int e^x\sin xdx\)
よって、
\(\displaystyle 2\int e^x\sin xdx=e^x\sin x-e^x\cos x\)
\(\displaystyle \int e^x\sin xdx=\frac{e^x(\sin x-\cos x)}{2}+C\)

\(\displaystyle =x\log(1+x^2)-2\int\frac{x^2}{1+x^2}dx\)
\(\displaystyle =x\log(1+x^2)-2\int\left(1-\frac{1}{1+x^2}\right)dx\)
\(\displaystyle =x\log(1+x^2)-2(x-\tan^{-1}x)+C\)

\(\displaystyle =\frac{x^2}{2}\tan^{-1}x-\int\frac{x^2}{2}・\frac{1}{1+x^2}dx\)
\(\displaystyle =\frac{x^2}{2}\tan^{-1}x-\frac{1}{2}\int\left(1-\frac{1}{1+x^2}\right)dx\)
\(\displaystyle =\frac{x^2}{2}\tan^{-1}x-\frac{1}{2}(x-\tan^{-1}x)+C\)
\(\displaystyle =\frac{x^2+1}{2}\tan^{-1}x-\frac{x}{2}+C\)

\(\displaystyle =\frac{x^{n+1}\log x}{n+1}-\frac{1}{n+1}\int x^ndx\)
\(\displaystyle =\frac{x^{n+1}\log x}{n+1}-\frac{1}{n+1}・\frac{x^{n+1}}{n+1}+C\)
\(\displaystyle =\frac{x^{n+1}}{n+1}\left(\log x-\frac{1}{n+1}\right)+C\)

\(\displaystyle =-x^3\cos x+3\int x^2\cos xdx\)
\(\displaystyle =-x^3\cos x+3(x^2\sin x-2\int x\sin xdx)\)
\(\displaystyle =-x^3\cos x+3x^2\sin x-6(-x\cos x+\int \cos xdx)\)
\(\displaystyle =-x^3\cos x+3x^2\sin x+6x\cos x-6\sin x+C\)

\(\displaystyle =x\cosh x-\int\cosh xdx\)
\(\displaystyle =x\cosh x-\sinh x+C\)

\(t=\sqrt{x-1}\)とおくと、\(\displaystyle dx=2tdt\)
\(\displaystyle =2\int_0^22・2tdt\)
\(\displaystyle =4\int_0^2t^2dt\)
\(\displaystyle =\frac{4}{3}[t^3]_0^2\)
\(\displaystyle =\frac{32}{3}\)

\(\displaystyle =\int_1^2(2t^{-3}-t^{-2})dt\)
\(\displaystyle =\left[-t^{-2}+t^{-1}\right]_1^2\)
\(\displaystyle =-\frac{1}{4}+\frac{1}{2}-(-1+1)\)
\(\displaystyle =\frac{1}{4}\)

\(\displaystyle =[\sin x]_0^\frac{\pi}{2}\)
\(\displaystyle =\sin\frac{\pi}{2}-\sin0\)
\(=1\)

\(t=-x^2\)とおくと、\(\displaystyle dx=-\frac{1}{2x}dt\)
\(\displaystyle =\int_0^{-1}e^t・-\frac{1}{2}dt\)
\(\displaystyle =\frac{1}{2}\int_{-1}^0e^tdt\)
\(\displaystyle =\frac{1}{2}[e^t]_{-1}^0\)
\(\displaystyle =\frac{1-e^{-1}}{2}\)

\(t=e^x+1\)とおくと、\(\displaystyle dx=\frac{1}{e^x}dt\)
\(\displaystyle =\int_2^3\frac{1}{t}dt\)
\(\displaystyle =[\log|t|]_2^3\)
\(\displaystyle =\log3-\log2\)
\(\displaystyle =\log\frac{3}{2}\)

\(t=\cos x\)とおくと、\(\displaystyle dx=-\frac{1}{\sin x}dt\)
\(\displaystyle =\int_1^0t^4・-dt\)
\(\displaystyle =\int_0^1t^4dt\)
\(\displaystyle =[\frac{t^5}{5}]_0^1\)
\(\displaystyle =\frac{1}{5}\)

\(t=\sqrt{1+\cos x}\)とおくと、\(\displaystyle dx=-\frac{2t}{\sin x}dt\)
\(\displaystyle =\int_\sqrt{2}^1\frac{-2t}{t}dt\)
\(\displaystyle =[-2t]_\sqrt{2}^1\)
\(\displaystyle =-2(1-\sqrt{2})\)
\(\displaystyle =2(\sqrt{2}-1)\)

\(\displaystyle =\int_0^{\frac{\pi}{2}}\left(\frac{1-\cos2x}{2}\right)^2dx\)
\(\displaystyle =\int_0^{\frac{\pi}{2}}\left(\frac{1-2\cos2x+\cos^22x}{4}\right)dx\)
\(\displaystyle =\int_0^{\frac{\pi}{2}}\left(\frac{1-2\cos2x+\frac{1+\cos4x}{2}}{4}\right)dx\)
\(\displaystyle =\int_0^{\frac{\pi}{2}}\left(\frac{3}{8}-\frac{1}{2}\cos2x+\frac{1}{8}\cos4x\right)dx\)
\(\displaystyle =\left[\frac{3}{8}x-\frac{\sin2x}{4}+\frac{\sin4x}{32}\right]_0^\sqrt{2}\)
\(\displaystyle =\frac{3}{8}・\frac{\pi}{2}\)
\(\displaystyle =\frac{3\pi}{16}\)

\(x^2,\cos x\)共に偶関数であるので、\(x^2\cos x\)は偶関数
\(\displaystyle =2\int_0^1x^2\cos xdx\)
\(\displaystyle =2(x^2\sin x-\int_0^12x\sin xdx)\)
\(\displaystyle =2([x^2\sin x]_0^1-[-2x\cos x]_0^1-[2\sin x]_0^1\)
\(\displaystyle =2(\sin1+2\cos1-2\sin1)\)
\(\displaystyle =2(2\cos1-\sin1)\)

\(\displaystyle =\left[\frac{1}{n}\sin nx\right]_0^\pi\)
\(\displaystyle =0\)

\(x=2\sin\theta\)とおくと、\(dx=2\cos\theta d\theta\)
\(\displaystyle =\int_0^{\frac{\pi}{6}}\frac{(2\sin\theta)^2}{2\cos\theta}2\cos\theta d\theta\)
\(\displaystyle =\int_0^{\frac{\pi}{6}}4\sin^2\theta d\theta\)
\(\displaystyle =\int_0^{\frac{\pi}{6}}2(1-\cos2\theta)d\theta\)
\(\displaystyle =2\left[\theta-\frac{\sin2\theta}{2}\right]_0^{\frac{\pi}{6}}\)
\(\displaystyle =2(\frac{\pi}{6}-\frac{\sqrt{3}}{4})\)
\(\displaystyle =\frac{\pi}{3}-\frac{\sqrt{3}}{2}\)

\(\displaystyle =[xe^x]_0^1-\int_0^1e^xdx\)
\(=e-[e^x]_0^1\)
\(=e-(e-1)\)
\(=1\)