(1) \(\displaystyle\int\cos{x}dx\)
(2) \(\displaystyle\int\cos^2{x}dx\)
(3) \(\displaystyle\int\cos^3{x}dx\)
(4) \(\displaystyle\int\cos^4{x}dx\)
2乗，3乗，4乗で計算の仕方が異なるよ．確認しておこう
【解答】\(C\) は積分定数
(1) \(\displaystyle\int\cos{x}dx＝\sin{x}+C\)
(2) \(\displaystyle\int\cos^2{x}dx\)
\(\displaystyle＝\int\frac{1}{2}(1+\cos{2x})dx\)
\(\displaystyle＝\frac{1}{2}(x+\frac{1}{2}\sin{2x})＋C\)
\(\displaystyle＝\frac{1}{2}x+\frac{1}{4}\sin{2x}＋C\)
(3) \(\displaystyle\int\cos^3{x}dx\)
\(\displaystyle＝\int\cos^2{x}・\cos{x}dx\)
\(\displaystyle＝\int(1-\sin^2{x})・(\sin{x})’dx\)
\(\displaystyle＝\int\lbrace(\sin{x})’-\sin^2{x}(\sin{x})’\rbrace dx\)
\(\displaystyle＝\sin{x}-\frac{1}{3}\sin^3{x}＋C\)
(4) \(\displaystyle\int\cos^4{x}dx\)
\(\displaystyle＝\int(\cos^2{x})^2dx\)
\(\displaystyle＝\int\lbrace\frac{1}{2}(1+\cos{2x})\rbrace ^2dx\)
ここで括弧の中を展開します
\(\displaystyle＝\int\frac{1}{4}(1+2\cos{2x}+\cos^2{2x})dx\)
\(\displaystyle＝\int\frac{1}{4}(1+2\cos{2x}+\frac{1+\cos{4x}}{2})dx\)
\(\displaystyle＝\frac{1}{4}\lbrace x+\sin{2x}+\frac{1}{2}(x+\frac{1}{4}\sin{4x})+C\rbrace\)
積分定数に数をかけても積分定数は \(\displaystyle C\) のままです
\(\displaystyle＝\frac{3}{8}x+\frac{1}{4}\sin{2x}+\frac{1}{32}\sin{4x}+C\)
