7-10-x-2-dx-x-2-3x-2-

Question Number 11864 by ankhaaankhaa last updated on 03/Apr/17

\underset{7}{\int^{10}} x^{2} d x / x^{2} - 3 x + 2 =

Answered by ajfour last updated on 03/Apr/17

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

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

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

= x + 4 \ln ∣ x - 2 ∣ - \ln ∣ x - 1 ∣ + C

\int_{7}^{10} \frac{x^{2}}{x^{2} - 3 x + 2} d x

= (7 - 3) + 4 \ln (\frac{10 - 2}{7 - 2}) - \ln (\frac{10 - 1}{7 - 1})

= 4 + 4 \ln (8 / 5) - \ln (3 / 2) .

Answered by sma3l2996 last updated on 03/Apr/17

A = \int_{7}^{10} \frac{x^{2} d x}{x^{2} - 3 x + 2} = \int_{7}^{10} (1 + \frac{3 x - 2}{x^{2} - 3 x + 2}) d x

= {[x]}_{7}^{10} + \frac{3}{2} \int_{7}^{10} \frac{2 x - \frac{4}{3}}{x^{2} - 3 x + 2} d x

= 3 + \frac{3}{2} \int_{7}^{10} \frac{2 x - 3}{x^{2} - 3 x + 2} d x + \frac{5}{2} \int_{7}^{10} \frac{d x}{(x - 1) (x - 2)}

= 3 + \frac{3}{2} {[l n ∣ x^{2} - 3 x + 2 ∣]}_{7}^{10} + \frac{5}{2} \int_{7}^{10} (\frac{1}{x - 2} - \frac{1}{x - 1}) d x

= 3 + \frac{3}{2} l n (\frac{36}{15}) + \frac{5}{2} {[l n ∣ \frac{x - 2}{x - 1} ∣]}_{7}^{10}

= \frac{3}{2} l n (\frac{12}{5}) + \frac{5}{2} (l n (\frac{8}{9}) - l n (\frac{5}{6})) + 3

A = \frac{3}{2} l n (\frac{12}{5}) + \frac{5}{2} l n (\frac{16}{15}) + 3