integral of 4(cos(x)+sin(2x))/(sin(x))

解

\int 4 \frac{cos ( x ) + sin ( 2 x )}{sin ( x )} d x

4 (ln ∣ sin (x) ∣ + 2 sin (x)) + C

解答ステップ

\int 4 \cdot \frac{cos ( x ) + sin ( 2 x )}{sin ( x )} d x

定数を除く:

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

= 4 \cdot \int \frac{cos ( x ) + sin ( 2 x )}{sin ( x )} d x

拡張

\frac{cos ( x ) + sin ( 2 x )}{sin ( x )} : \frac{cos ( x )}{sin ( x )} + \frac{sin ( 2 x )}{sin ( x )}

= 4 \cdot \int \frac{cos ( x )}{sin ( x )} + \frac{sin ( 2 x )}{sin ( x )} 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

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

\int \frac{cos ( x )}{sin ( x )} d x = ln ∣ sin (x) ∣

\int \frac{sin ( 2 x )}{sin ( x )} d x = 2 sin (x)

= 4 (ln ∣ sin (x) ∣ + 2 sin (x))

定数を解答に追加する

= 4 (ln ∣ sin (x) ∣ + 2 sin (x)) + C