integral of (sin(x))/(sin(x)+2cos(x))

解

\int \frac{sin ( x )}{sin ( x ) + 2 cos ( x )} d x

- 2 (\frac{1}{5} ln ∣ tan (x) + 2 ∣ + \frac{1}{5} (- \frac{1}{2} ln (sec^{2} (x)) + 2 arctan (tan (x)))) + x + C

解答ステップ

\int \frac{sin ( x )}{sin ( x ) + 2 cos ( x )} d x

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

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

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{2 cos ( x )}{sin ( x ) + 2 cos ( x )} d x + \int 1 d x

\int \frac{2 cos ( x )}{sin ( x ) + 2 cos ( x )} d x = 2 (\frac{1}{5} ln ∣ tan (x) + 2 ∣ + \frac{1}{5} (- \frac{1}{2} ln (sec^{2} (x)) + 2 arctan (tan (x))))

\int 1 d x = x

= - 2 (\frac{1}{5} ln ∣ tan (x) + 2 ∣ + \frac{1}{5} (- \frac{1}{2} ln (sec^{2} (x)) + 2 arctan (tan (x)))) + x

定数を解答に追加する

= - 2 (\frac{1}{5} ln ∣ tan (x) + 2 ∣ + \frac{1}{5} (- \frac{1}{2} ln (sec^{2} (x)) + 2 arctan (tan (x)))) + x + C