integral of xcos^2(5x)

解

\int x cos^{2} (5 x) d x

\frac{1}{200} (50 x^{2} + 10 x sin (10 x) + cos (10 x)) + C

解答ステップ

\int x cos^{2} (5 x) d x

三角関数の公式を使用して書き換える

= \int \frac{1}{2} x (1 + cos (10 x)) d x

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int x (1 + cos (10 x)) d x

部分積分を適用する

= \frac{1}{2} (x (x + \frac{1}{10} sin (10 x)) - \int x + \frac{1}{10} sin (10 x) d x)

\int x + \frac{1}{10} sin (10 x) d x = \frac{x ^{2}}{2} - \frac{1}{100} cos (10 x)

= \frac{1}{2} (x (x + \frac{1}{10} sin (10 x)) - (\frac{x ^{2}}{2} - \frac{1}{100} cos (10 x)))

簡素化

\frac{1}{2} (x (x + \frac{1}{10} sin (10 x)) - (\frac{x ^{2}}{2} - \frac{1}{100} cos (10 x))) : \frac{1}{200} (50 x^{2} + 10 x sin (10 x) + cos (10 x))

= \frac{1}{200} (50 x^{2} + 10 x sin (10 x) + cos (10 x))

定数を解答に追加する

= \frac{1}{200} (50 x^{2} + 10 x sin (10 x) + cos (10 x)) + C