What is the integral of x/(2x^2-8x+3) ?

The integral of x/(2x^2-8x+3) is

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integral of x/(2x^2-8x+3)

\int \frac{x}{2 x ^{2} - 8 x + 3} d x

\frac{1}{2 10} x ln \frac{2}{5} (x - 2) - 1 - \frac{1}{2 10} x ln \frac{2}{5} x - 2 \frac{2}{5} - 1 + \frac{10 - 4}{4 10} ln \frac{2}{5} (x - 2) + 1 + \frac{10 + 4}{4 10} ln \frac{2}{5} x - 2 \frac{2}{5} - 1 - \frac{1}{2} + C

Solution steps

\int \frac{x}{2 x ^{2} - 8 x + 3} d x

Apply Integration By Parts

: - \frac{1}{2 10} x (ln \frac{2}{5} (x - 2) + 1 - ln \frac{2}{5} (x - 2) - 1) - \int - \frac{1}{2 10} (ln \frac{2}{5} (x - 2) + 1 - ln \frac{2}{5} (x - 2) - 1) d x

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

\int - \frac{1}{2 10} (ln \frac{2}{5} (x - 2) + 1 - ln \frac{2}{5} (x - 2) - 1) d x = - \frac{1}{4 5} (2 x ln \frac{2}{5} (x - 2) + 1 - 2 x ln \frac{2}{5} x - 2 \frac{2}{5} - 1 + 22 ln \frac{2}{5} x - 2 \frac{2}{5} - 1 + 5 ln \frac{2}{5} (x - 2) + 1 + 5 ln \frac{2}{5} x - 2 \frac{2}{5} - 1 - 22 ln \frac{2}{5} (x - 2) + 1 - 25)

= - \frac{1}{2 10} x (ln \frac{2}{5} (x - 2) + 1 - ln \frac{2}{5} (x - 2) - 1) - (- \frac{1}{4 5} (2 x ln \frac{2}{5} (x - 2) + 1 - 2 x ln \frac{2}{5} x - 2 \frac{2}{5} - 1 + 22 ln \frac{2}{5} x - 2 \frac{2}{5} - 1 + 5 ln \frac{2}{5} (x - 2) + 1 + 5 ln \frac{2}{5} x - 2 \frac{2}{5} - 1 - 22 ln \frac{2}{5} (x - 2) + 1 - 25))

Simplify

- \frac{1}{2 10} x (ln \frac{2}{5} (x - 2) + 1 - ln \frac{2}{5} (x - 2) - 1) - (- \frac{1}{4 5} (2 x ln \frac{2}{5} (x - 2) + 1 - 2 x ln \frac{2}{5} x - 2 \frac{2}{5} - 1 + 22 ln \frac{2}{5} x - 2 \frac{2}{5} - 1 + 5 ln \frac{2}{5} (x - 2) + 1 + 5 ln \frac{2}{5} x - 2 \frac{2}{5} - 1 - 22 ln \frac{2}{5} (x - 2) + 1 - 25)) : \frac{1}{2 10} x ln \frac{2}{5} (x - 2) - 1 - \frac{1}{2 10} x ln \frac{2}{5} x - 2 \frac{2}{5} - 1 + \frac{10 - 4}{4 10} ln \frac{2}{5} (x - 2) + 1 + \frac{10 + 4}{4 10} ln \frac{2}{5} x - 2 \frac{2}{5} - 1 - \frac{1}{2}

= \frac{1}{2 10} x ln \frac{2}{5} (x - 2) - 1 - \frac{1}{2 10} x ln \frac{2}{5} x - 2 \frac{2}{5} - 1 + \frac{10 - 4}{4 10} ln \frac{2}{5} (x - 2) + 1 + \frac{10 + 4}{4 10} ln \frac{2}{5} x - 2 \frac{2}{5} - 1 - \frac{1}{2}

Add a constant to the solution

= \frac{1}{2 10} x ln \frac{2}{5} (x - 2) - 1 - \frac{1}{2 10} x ln \frac{2}{5} x - 2 \frac{2}{5} - 1 + \frac{10 - 4}{4 10} ln \frac{2}{5} (x - 2) + 1 + \frac{10 + 4}{4 10} ln \frac{2}{5} x - 2 \frac{2}{5} - 1 - \frac{1}{2} + C

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