What is the integral from 2 to 3 of (1000x)/(x^2+e) ?

The integral from 2 to 3 of (1000x)/(x^2+e) is 500(ln(9+e)-ln(4+e))

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integral from 2 to 3 of (1000x)/(x^2+e)

\int_{2}^{3} \frac{1000 x}{x ^{2} + e} d x

500 (ln (9 + e) - ln (4 + e))

Decimal

278.15886 \dots

Solution steps

\int_{2}^{3} \frac{1000 x}{x ^{2} + e} d x

Take the constant out:

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

= 1000 \cdot \int_{2}^{3} \frac{x}{x ^{2} + e} d x

Apply u-substitution

: \int_{4 + e}^{9 + e} \frac{1}{2 u} d u

= 1000 \cdot \int_{4 + e}^{9 + e} \frac{1}{2 u} d u

Take the constant out:

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

= 1000 \cdot \frac{1}{2} \cdot \int_{4 + e}^{9 + e} \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 1000 \cdot \frac{1}{2} [ln ∣ u ∣]_{4 + e}^{9 + e}

Simplify

1000 \cdot \frac{1}{2} [ln ∣ u ∣]_{4 + e}^{9 + e} : 500 [ln ∣ u ∣]_{4 + e}^{9 + e}

= 500 [ln ∣ u ∣]_{4 + e}^{9 + e}

Compute the boundaries:

ln (9 + e) - ln (4 + e)

= 500 (ln (9 + e) - ln (4 + e))

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What is the integral from 2 to 3 of (1000x)/(x^2+e) ?
The integral from 2 to 3 of (1000x)/(x^2+e) is 500(ln(9+e)-ln(4+e))