integral of cos^2(x)-cos^4(x)

解

\int cos^{2} (x) - cos^{4} (x) d x

x cos^{2} (x) sin^{2} (x) - \frac{1}{2} (- \frac{1}{4} x cos (4 x) + \frac{1}{16} sin (4 x)) + C

解答ステップ

\int cos^{2} (x) - cos^{4} (x) d x

部分積分を適用する

= x cos^{2} (x) sin^{2} (x) - \int \frac{1}{2} x sin (4 x) d x

\int \frac{1}{2} x sin (4 x) d x = \frac{1}{2} (- \frac{1}{4} x cos (4 x) + \frac{1}{16} sin (4 x))

= x cos^{2} (x) sin^{2} (x) - \frac{1}{2} (- \frac{1}{4} x cos (4 x) + \frac{1}{16} sin (4 x))

定数を解答に追加する

= x cos^{2} (x) sin^{2} (x) - \frac{1}{2} (- \frac{1}{4} x cos (4 x) + \frac{1}{16} sin (4 x)) + C