integral of (3x-1)/(x^2+10x+28)

解答

\int \frac{3 x - 1}{x ^{2} + 10 x + 28} d x

3 (\frac{1}{2} ln x^{2} + 10 x + 28 - \frac{5}{3} arctan (\frac{1}{3} (x + 5))) - \frac{1}{3} arctan (\frac{x + 5}{3}) + C

求解步骤

\int \frac{3 x - 1}{x ^{2} + 10 x + 28} d x

乘开

\frac{3 x - 1}{x ^{2} + 10 x + 28} : \frac{3 x}{x ^{2} + 10 x + 28} - \frac{1}{x ^{2} + 10 x + 28}

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{3 x}{x ^{2} + 10 x + 28} d x - \int \frac{1}{x ^{2} + 10 x + 28} d x

\int \frac{3 x}{x ^{2} + 10 x + 28} d x = 3 (\frac{1}{2} ln x^{2} + 10 x + 28 - \frac{5}{3} arctan (\frac{1}{3} (x + 5)))

\int \frac{1}{x ^{2} + 10 x + 28} d x = \frac{1}{3} arctan (\frac{x + 5}{3})

= 3 (\frac{1}{2} ln x^{2} + 10 x + 28 - \frac{5}{3} arctan (\frac{1}{3} (x + 5))) - \frac{1}{3} arctan (\frac{x + 5}{3})

解答补常数

= 3 (\frac{1}{2} ln x^{2} + 10 x + 28 - \frac{5}{3} arctan (\frac{1}{3} (x + 5))) - \frac{1}{3} arctan (\frac{x + 5}{3}) + C

d er i v a t i v e x^{2} sin^{6} (x) + x cos^{- 3} (x)

\frac{d}{d x} (\frac{e ^{2 x}}{e ^{2 x} + 1})

\frac{d}{d y} (\frac{y}{y + \frac{c}{y}})

\int - \frac{7}{x ^{2} + 16} d x

t an g e n t f (x) = \frac{1}{2 x}, at x = 64