What is the integral from 3 to 5 of (3x)/(2x^2-6) ?

The integral from 3 to 5 of (3x)/(2x^2-6) is 3/4 ln(11/3)

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integral from 3 to 5 of (3x)/(2x^2-6)

\int_{3}^{5} \frac{3 x}{2 x ^{2} - 6} d x

\frac{3}{4} ln (\frac{11}{3})

Decimal Notation

0.97446 \dots

Solution steps

\int_{3}^{5} \frac{3 x}{2 x ^{2} - 6} d x

Take the constant out:

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

= 3 \cdot \int_{3}^{5} \frac{x}{2 x ^{2} - 6} d x

Apply u-substitution

: \int_{12}^{44} \frac{1}{4 u} d u

= 3 \cdot \int_{12}^{44} \frac{1}{4 u} d u

Take the constant out:

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

= 3 \cdot \frac{1}{4} \cdot \int_{12}^{44} \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 3 \cdot \frac{1}{4} [ln ∣ u ∣]_{12}^{44}

Simplify

3 \cdot \frac{1}{4} [ln ∣ u ∣]_{12}^{44} : \frac{3}{4} [ln ∣ u ∣]_{12}^{44}

= \frac{3}{4} [ln ∣ u ∣]_{12}^{44}

Compute the boundaries:

ln (44) - ln (12)

= \frac{3}{4} (ln (44) - ln (12))

Simplify

= \frac{3}{4} ln (\frac{11}{3})

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What is the integral from 3 to 5 of (3x)/(2x^2-6) ?
The integral from 3 to 5 of (3x)/(2x^2-6) is 3/4 ln(11/3)