What is the integral from 50 to 100 of x/(20+3x^2) ?

The integral from 50 to 100 of x/(20+3x^2) is 1/6 ln(1501/376)

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integral from 50 to 100 of x/(20+3x^2)

\int_{50}^{100} \frac{x}{20 + 3 x ^{2}} d x

\frac{1}{6} ln (\frac{1501}{376})

Decimal

0.23071 \dots

Solution steps

\int_{50}^{100} \frac{x}{20 + 3 x ^{2}} d x

Apply u-substitution

: \int_{7520}^{30020} \frac{1}{6 u} d u

= \int_{7520}^{30020} \frac{1}{6 u} d u

Take the constant out:

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

= \frac{1}{6} \cdot \int_{7520}^{30020} \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{6} [ln ∣ u ∣]_{7520}^{30020}

Compute the boundaries:

ln (30020) - ln (7520)

= \frac{1}{6} (ln (30020) - ln (7520))

Simplify

= \frac{1}{6} ln (\frac{1501}{376})

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What is the integral from 50 to 100 of x/(20+3x^2) ?
The integral from 50 to 100 of x/(20+3x^2) is 1/6 ln(1501/376)