integral of cos^2(ln(b))

解

\int cos^{2} (ln (b)) d b

\frac{1}{10} (5 b + 2 b sin (2 ln (b)) + b cos (2 ln (b))) + C

解答ステップ

\int cos^{2} (ln (b)) d b

三角関数の公式を使用して書き換える

= \int \frac{1 + cos ( 2 ln ( b ) )}{2} d b

定数を除く:

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

= \frac{1}{2} \cdot \int 1 + cos (2 ln (b)) d b

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} (\int 1 d b + \int cos (2 ln (b)) d b)

\int 1 d b = b

\int cos (2 ln (b)) d b = \frac{2}{5} b sin (2 ln (b)) + \frac{1}{5} b cos (2 ln (b))

= \frac{1}{2} (b + \frac{2}{5} b sin (2 ln (b)) + \frac{1}{5} b cos (2 ln (b)))

簡素化

\frac{1}{2} (b + \frac{2}{5} b sin (2 ln (b)) + \frac{1}{5} b cos (2 ln (b))) : \frac{1}{10} (5 b + 2 b sin (2 ln (b)) + b cos (2 ln (b)))

= \frac{1}{10} (5 b + 2 b sin (2 ln (b)) + b cos (2 ln (b)))

定数を解答に追加する

= \frac{1}{10} (5 b + 2 b sin (2 ln (b)) + b cos (2 ln (b))) + C