integral of ((7x+1))/(x^2-7x+6)

解答

\int \frac{( 7 x + 1 )}{x ^{2} - 7 x + 6} d x

14 (\frac{1}{4} ln 4 x^{2} - 28 x + 24 - \frac{7}{20} (ln - \frac{2}{5} + \frac{2}{5} x - ln - \frac{12}{5} + \frac{2}{5} x)) - \frac{1}{5} ln ∣ x - 1 ∣ + \frac{1}{5} ln ∣ x - 6 ∣ + C

求解步骤

\int \frac{7 x + 1}{x ^{2} - 7 x + 6} d x

乘开

\frac{7 x + 1}{x ^{2} - 7 x + 6} : \frac{7 x}{x ^{2} - 7 x + 6} + \frac{1}{x ^{2} - 7 x + 6}

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{7 x}{x ^{2} - 7 x + 6} d x + \int \frac{1}{x ^{2} - 7 x + 6} d x

\int \frac{7 x}{x ^{2} - 7 x + 6} d x = 14 (\frac{1}{4} ln 4 x^{2} - 28 x + 24 - \frac{7}{20} (ln - \frac{2}{5} + \frac{2}{5} x - ln - \frac{12}{5} + \frac{2}{5} x))

\int \frac{1}{x ^{2} - 7 x + 6} d x = - \frac{1}{5} ln ∣ x - 1 ∣ + \frac{1}{5} ln ∣ x - 6 ∣

= 14 (\frac{1}{4} ln 4 x^{2} - 28 x + 24 - \frac{7}{20} (ln - \frac{2}{5} + \frac{2}{5} x - ln - \frac{12}{5} + \frac{2}{5} x)) - \frac{1}{5} ln ∣ x - 1 ∣ + \frac{1}{5} ln ∣ x - 6 ∣

解答补常数

= 14 (\frac{1}{4} ln 4 x^{2} - 28 x + 24 - \frac{7}{20} (ln - \frac{2}{5} + \frac{2}{5} x - ln - \frac{12}{5} + \frac{2}{5} x)) - \frac{1}{5} ln ∣ x - 1 ∣ + \frac{1}{5} ln ∣ x - 6 ∣ + C

\int (sec^{2} (x) tan^{3} (x)) d x

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

\int \frac{( 2 x - 3 )}{x ^{2} + 6 x + 13} d x

\int \frac{x + 3}{( x ^{2} + 2 x + 5 ) ^{2}} d x

\int \frac{t ^{3}}{t ^{2} - 9} d t