integral of (4x+2)/(sqrt(x^2+5x-1))

解答

\int \frac{4 x + 2}{x ^{2} + 5 x - 1} d x

4 (x^{2} + 5 x - 1 - \frac{5}{2} ln \frac{1}{29} (4 x^{2} + 20 x - 4) + \frac{2}{29} x + \frac{5}{29}) + 2 ln \frac{1}{29} (4 x^{2} + 20 x - 4) + \frac{1}{29} (2 x + 5) + C

求解步骤

\int \frac{4 x + 2}{x ^{2} + 5 x - 1} d x

乘开

\frac{4 x + 2}{x ^{2} + 5 x - 1} : \frac{4 x}{x ^{2} + 5 x - 1} + \frac{2}{x ^{2} + 5 x - 1}

= \int \frac{4 x}{x ^{2} + 5 x - 1} + \frac{2}{x ^{2} + 5 x - 1} 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 \frac{4 x}{x ^{2} + 5 x - 1} d x + \int \frac{2}{x ^{2} + 5 x - 1} d x

\int \frac{4 x}{x ^{2} + 5 x - 1} d x = 4 (x^{2} + 5 x - 1 - \frac{5}{2} ln \frac{4 x ^{2} + 20 x - 4}{29} + \frac{2}{29} x + \frac{5}{29})

\int \frac{2}{x ^{2} + 5 x - 1} d x = 2 ln \frac{1}{29} (4 x^{2} + 20 x - 4) + \frac{2}{29} x + \frac{5}{29}

= 4 (x^{2} + 5 x - 1 - \frac{5}{2} ln \frac{4 x ^{2} + 20 x - 4}{29} + \frac{2}{29} x + \frac{5}{29}) + 2 ln \frac{1}{29} (4 x^{2} + 20 x - 4) + \frac{2}{29} x + \frac{5}{29}

化简

4 (x^{2} + 5 x - 1 - \frac{5}{2} ln \frac{4 x ^{2} + 20 x - 4}{29} + \frac{2}{29} x + \frac{5}{29}) + 2 ln \frac{1}{29} (4 x^{2} + 20 x - 4) + \frac{2}{29} x + \frac{5}{29} : 4 (x^{2} + 5 x - 1 - \frac{5}{2} ln \frac{1}{29} (4 x^{2} + 20 x - 4) + \frac{2}{29} x + \frac{5}{29}) + 2 ln \frac{1}{29} (4 x^{2} + 20 x - 4) + \frac{1}{29} (2 x + 5)

= 4 (x^{2} + 5 x - 1 - \frac{5}{2} ln \frac{1}{29} (4 x^{2} + 20 x - 4) + \frac{2}{29} x + \frac{5}{29}) + 2 ln \frac{1}{29} (4 x^{2} + 20 x - 4) + \frac{1}{29} (2 x + 5)

解答补常数

= 4 (x^{2} + 5 x - 1 - \frac{5}{2} ln \frac{1}{29} (4 x^{2} + 20 x - 4) + \frac{2}{29} x + \frac{5}{29}) + 2 ln \frac{1}{29} (4 x^{2} + 20 x - 4) + \frac{1}{29} (2 x + 5) + C

\int \frac{sin ( 3 x )}{cos ^{2} ( 3 x ) + 6} d x

\int - 8.3 x + 5.1 d x

\int x cos (3 x) - sin (3 x) d x

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

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