integral of (cos(θ))/(5-3cos(θ))

解

\int \frac{cos ( θ )}{5 - 3 cos ( θ )} d θ

- \frac{1}{3} θ + \frac{5}{6} arctan (2 tan (\frac{θ}{2})) + C

解答ステップ

\int \frac{cos ( θ )}{5 - 3 cos ( θ )} d θ

長除法

\frac{cos ( θ )}{5 - 3 cos ( θ )} : - \frac{1}{3} + \frac{5}{3 ( - 3 cos ( θ ) + 5 )}

= \int - \frac{1}{3} + \frac{5}{3 ( - 3 cos ( θ ) + 5 )} d θ

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

= - \int \frac{1}{3} d θ + \int \frac{5}{3 ( - 3 cos ( θ ) + 5 )} d θ

\int \frac{1}{3} d θ = \frac{1}{3} θ

\int \frac{5}{3 ( - 3 cos ( θ ) + 5 )} d θ = \frac{5}{6} arctan (2 tan (\frac{θ}{2}))

= - \frac{1}{3} θ + \frac{5}{6} arctan (2 tan (\frac{θ}{2}))

定数を解答に追加する

= - \frac{1}{3} θ + \frac{5}{6} arctan (2 tan (\frac{θ}{2})) + C