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

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

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integral from-2 to 2 of (x/(x+3))

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

4 - 3 ln (5)

Decimal

- 0.82831 \dots

Solution steps

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

Apply u-substitution

: \int_{1}^{5} \frac{u - 3}{u} d u

= \int_{1}^{5} \frac{u - 3}{u} d u

Expand

\frac{u - 3}{u} : 1 - \frac{3}{u}

= \int_{1}^{5} 1 - \frac{3}{u} d u

Apply the Sum Rule:

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

= \int_{1}^{5} 1 d u - \int_{1}^{5} \frac{3}{u} d u

\int_{1}^{5} 1 d u = 4

\int_{1}^{5} \frac{3}{u} d u = 3 ln (5)

= 4 - 3 ln (5)

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