What is the integral of (3x)/(2x^2-9x-5) ?

The integral of (3x)/(2x^2-9x-5) is

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integral of (3x)/(2x^2-9x-5)

\int \frac{3 x}{2 x ^{2} - 9 x - 5} d x

3 (- \frac{1}{11} x (ln \frac{2}{11} + \frac{4}{11} x - ln - \frac{20}{11} + \frac{4}{11} x) + \frac{1}{22} (2 x (ln \frac{2}{11} + \frac{4}{11} x - ln - \frac{20}{11} + \frac{4}{11} x) + ln ∣ 2 x + 1 ∣ + 10 ln ∣ 2 x - 10 ∣)) + C

Solution steps

\int \frac{3 x}{2 x ^{2} - 9 x - 5} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \int \frac{x}{2 x ^{2} - 9 x - 5} d x

Apply Integration By Parts

: - \frac{1}{11} x (ln \frac{2}{11} + \frac{4}{11} x - ln - \frac{20}{11} + \frac{4}{11} x) - \int - \frac{1}{11} (ln \frac{2}{11} + \frac{4}{11} x - ln - \frac{20}{11} + \frac{4}{11} x) d x

= 3 (- \frac{1}{11} x (ln \frac{2}{11} + \frac{4}{11} x - ln - \frac{20}{11} + \frac{4}{11} x) - \int - \frac{1}{11} (ln \frac{2}{11} + \frac{4}{11} x - ln - \frac{20}{11} + \frac{4}{11} x) d x)

\int - \frac{1}{11} (ln \frac{2}{11} + \frac{4}{11} x - ln - \frac{20}{11} + \frac{4}{11} x) d x = - \frac{1}{22} (2 x (ln \frac{2}{11} + \frac{4}{11} x - ln - \frac{20}{11} + \frac{4}{11} x) + ln ∣ 2 x + 1 ∣ + 10 ln ∣ 2 x - 10 ∣)

= 3 (- \frac{1}{11} x (ln \frac{2}{11} + \frac{4}{11} x - ln - \frac{20}{11} + \frac{4}{11} x) - (- \frac{1}{22} (2 x (ln \frac{2}{11} + \frac{4}{11} x - ln - \frac{20}{11} + \frac{4}{11} x) + ln ∣ 2 x + 1 ∣ + 10 ln ∣ 2 x - 10 ∣)))

Simplify

3 (- \frac{1}{11} x (ln \frac{2}{11} + \frac{4}{11} x - ln - \frac{20}{11} + \frac{4}{11} x) - (- \frac{1}{22} (2 x (ln \frac{2}{11} + \frac{4}{11} x - ln - \frac{20}{11} + \frac{4}{11} x) + ln ∣ 2 x + 1 ∣ + 10 ln ∣ 2 x - 10 ∣))) : 3 (- \frac{1}{11} x (ln \frac{2}{11} + \frac{4}{11} x - ln - \frac{20}{11} + \frac{4}{11} x) + \frac{1}{22} (2 x (ln \frac{2}{11} + \frac{4}{11} x - ln - \frac{20}{11} + \frac{4}{11} x) + ln ∣ 2 x + 1 ∣ + 10 ln ∣ 2 x - 10 ∣))

= 3 (- \frac{1}{11} x (ln \frac{2}{11} + \frac{4}{11} x - ln - \frac{20}{11} + \frac{4}{11} x) + \frac{1}{22} (2 x (ln \frac{2}{11} + \frac{4}{11} x - ln - \frac{20}{11} + \frac{4}{11} x) + ln ∣ 2 x + 1 ∣ + 10 ln ∣ 2 x - 10 ∣))

Add a constant to the solution

= 3 (- \frac{1}{11} x (ln \frac{2}{11} + \frac{4}{11} x - ln - \frac{20}{11} + \frac{4}{11} x) + \frac{1}{22} (2 x (ln \frac{2}{11} + \frac{4}{11} x - ln - \frac{20}{11} + \frac{4}{11} x) + ln ∣ 2 x + 1 ∣ + 10 ln ∣ 2 x - 10 ∣)) + C

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