integral of cos(x/2)*cos(nx)

解

\int cos (\frac{x}{2}) \cdot cos (n x) d x

\frac{1}{n} sin (n x) cos (\frac{x}{2}) + \frac{1}{4 n} (- \frac{2}{2 n + 1} sin (n x + \frac{x}{2}) + \frac{2}{2 n - 1} sin (n x - \frac{x}{2})) + C

解答ステップ

\int cos (\frac{x}{2}) cos (n x) d x

部分積分を適用する

= \frac{1}{n} sin (n x) cos (\frac{x}{2}) - \int - \frac{1}{2 n} sin (n x) sin (\frac{x}{2}) d x

\int - \frac{1}{2 n} sin (n x) sin (\frac{x}{2}) d x = - \frac{1}{4 n} (- \frac{2}{2 n + 1} sin (n x + \frac{x}{2}) + \frac{2}{2 n - 1} sin (n x - \frac{x}{2}))

= \frac{1}{n} sin (n x) cos (\frac{x}{2}) - (- \frac{1}{4 n} (- \frac{2}{2 n + 1} sin (n x + \frac{x}{2}) + \frac{2}{2 n - 1} sin (n x - \frac{x}{2})))

簡素化

= \frac{1}{n} sin (n x) cos (\frac{x}{2}) + \frac{1}{4 n} (- \frac{2}{2 n + 1} sin (n x + \frac{x}{2}) + \frac{2}{2 n - 1} sin (n x - \frac{x}{2}))

定数を解答に追加する

= \frac{1}{n} sin (n x) cos (\frac{x}{2}) + \frac{1}{4 n} (- \frac{2}{2 n + 1} sin (n x + \frac{x}{2}) + \frac{2}{2 n - 1} sin (n x - \frac{x}{2})) + C