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

解答

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

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

求解步骤

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

乘开

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

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

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

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

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

解答补常数

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

\int \frac{1 + x ^{2}}{2 x} d x

\int (x^{2} + x + 8) x + 2 d x

\int \frac{x ^{2}}{12 - 5 x ^{3}} d x

\int \frac{( 2 x ^{3} )}{x ^{2} + x} d x

\int 2 x \cdot \frac{1}{x} d x