What is the integral from 0 to 1 of x/(x^2+3x+2) ?

The integral from 0 to 1 of x/(x^2+3x+2) is -3ln(2)+2ln(3)

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integral from 0 to 1 of x/(x^2+3x+2)

\int_{0}^{1} \frac{x}{x ^{2} + 3 x + 2} d x

- 3 ln (2) + 2 ln (3)

Decimal

0.11778 \dots

Solution steps

\int_{0}^{1} \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_{0}^{1} - \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_{0}^{1} \frac{1}{x + 1} d x + \int_{0}^{1} \frac{2}{x + 2} d x

\int_{0}^{1} \frac{1}{x + 1} d x = ln (2)

\int_{0}^{1} \frac{2}{x + 2} d x = 2 (ln (3) - ln (2))

= - ln (2) + 2 (ln (3) - ln (2))

Simplify

- ln (2) + 2 (ln (3) - ln (2)) : - 3 ln (2) + 2 ln (3)

= - 3 ln (2) + 2 ln (3)

\int_{0}^{1} x^{3} - x d x

\int_{0}^{2} \frac{1}{1 + e ^{1 - x}} d x

\int_{0}^{\frac{π}{2}} x cos^{2} (x) d x

\int_{0}^{3} x^{2} - 1 d x

\int_{0}^{t} e^{t} d t

What is the integral from 0 to 1 of x/(x^2+3x+2) ?
The integral from 0 to 1 of x/(x^2+3x+2) is -3ln(2)+2ln(3)