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

The integral from-3 to 2 of (2x)/(x^2+5) is -ln(14)+2ln(3)

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integral from-3 to 2 of (2x)/(x^2+5)

\int_{- 3}^{2} \frac{2 x}{x ^{2} + 5} d x

- ln (14) + 2 ln (3)

Solution steps

\int_{- 3}^{2} \frac{2 x}{x ^{2} + 5} d x

Take the constant out:

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

= 2 \cdot \int_{- 3}^{2} \frac{x}{x ^{2} + 5} d x

Apply u-substitution

: \int_{14}^{9} \frac{1}{2 u} d u

= 2 \cdot \int_{14}^{9} \frac{1}{2 u} d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= 2 (- \int_{9}^{14} \frac{1}{2 u} d u)

Take the constant out:

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

= 2 (- \frac{1}{2} \cdot \int_{9}^{14} \frac{1}{u} d u)

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 (- \frac{1}{2} [ln ∣ u ∣]_{9}^{14})

Simplify

2 (- \frac{1}{2} [ln ∣ u ∣]_{9}^{14}) : - [ln ∣ u ∣]_{9}^{14}

= - [ln ∣ u ∣]_{9}^{14}

Compute the boundaries:

ln (14) - 2 ln (3)

= - (ln (14) - 2 ln (3))

Simplify

= - ln (14) + 2 ln (3)

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\int_{- π}^{π} cos (\frac{x}{2}) d x

What is the integral from-3 to 2 of (2x)/(x^2+5) ?
The integral from-3 to 2 of (2x)/(x^2+5) is -ln(14)+2ln(3)