integral of ((1+cos(x)))/(sin(x))

解答

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

ln ∣ sin (x) ∣ + ln tan (\frac{x}{2}) + C

求解步骤

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

使用三角恒等式改写

= \int (1 + cos (x)) csc (x) d x

化简

(1 + cos (x)) csc (x) : cot (x) + csc (x)

= \int cot (x) + csc (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

= \int cot (x) d x + \int csc (x) d x

\int cot (x) d x = ln ∣ sin (x) ∣

\int csc (x) d x = ln tan (\frac{x}{2})

= ln ∣ sin (x) ∣ + ln tan (\frac{x}{2})

解答补常数

= ln ∣ sin (x) ∣ + ln tan (\frac{x}{2}) + C