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

解答

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

x tan (x) + ln ∣ cos (x) ∣ - x sin (x) tan (x) - x cos (x) + ln ∣ tan (x) + sec (x) ∣ + \frac{2}{tan ( \frac{x}{2} ) + 1} + x + C

求解步骤

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

乘开

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

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

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

= x tan (x) + ln ∣ cos (x) ∣ - x sin (x) tan (x) - x cos (x) + ln ∣ tan (x) + sec (x) ∣ + \frac{2}{tan ( \frac{x}{2} ) + 1} + x

解答补常数

= x tan (x) + ln ∣ cos (x) ∣ - x sin (x) tan (x) - x cos (x) + ln ∣ tan (x) + sec (x) ∣ + \frac{2}{tan ( \frac{x}{2} ) + 1} + x + C

\int x^{2} (L - x) (2 L - x) d x

\int \frac{4 + 3 x}{x} d x

\int \frac{sin ( 2 x )}{19 + cos ^{2} ( x )} d x

\int (5 x - 2)^{9} d x

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