What is the integral from 2 to 4 of x/(x^2+3x+2) ?

The integral from 2 to 4 of x/(x^2+3x+2) is -ln(5)+ln(3)+2ln(3/2)

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integral from 2 to 4 of x/(x^2+3x+2)

\int_{2}^{4} \frac{x}{x ^{2} + 3 x + 2} d x

- ln (5) + ln (3) + 2 ln (\frac{3}{2})

Decimal

0.30010 \dots

Solution steps

\int_{2}^{4} \frac{x}{x ^{2} + 3 x + 2} d x

Take the partial fraction of

\frac{x}{x ^{2} + 3 x + 2} : - \frac{1}{x + 1} + \frac{2}{x + 2}

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int_{2}^{4} \frac{1}{x + 1} d x = ln (5) - ln (3)

\int_{2}^{4} \frac{2}{x + 2} d x = 2 ln (\frac{3}{2})

= - (ln (5) - ln (3)) + 2 ln (\frac{3}{2})

Simplify

= - ln (5) + ln (3) + 2 ln (\frac{3}{2})

What is the integral from 2 to 4 of x/(x^2+3x+2) ?
The integral from 2 to 4 of x/(x^2+3x+2) is -ln(5)+ln(3)+2ln(3/2)