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

解答

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

4 (ln ∣ sin (x) ∣ + 2 sin (x)) + C

求解步骤

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

提出常数:

\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

乘开

\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

使用积分加法定则:

\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))

解答补常数

= 4 (ln ∣ sin (x) ∣ + 2 sin (x)) + C

d er i v a t i v e y = cos^{2} (1 - 3 x^{2})

x \to 0 - lim (e^{\frac{6}{x}})

x \to - 1 + lim (arccos (x))

2 y + (x + 1) \frac{d y}{d x} = 3 x + 3

d er i v a t i v e \frac{480 t}{( t ^{2} + 5 ) ^{3}}