What is the integral from 1 to 5 of (5x)/(x^2+2) ?

The integral from 1 to 5 of (5x)/(x^2+2) is 5ln(3)

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integral from 1 to 5 of (5x)/(x^2+2)

\int_{1}^{5} \frac{5 x}{x ^{2} + 2} d x

5 ln (3)

Solution steps

\int_{1}^{5} \frac{5 x}{x ^{2} + 2} d x

Take the constant out:

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

= 5 \cdot \int_{1}^{5} \frac{x}{x ^{2} + 2} d x

Apply u-substitution

: \int_{3}^{27} \frac{1}{2 u} d u

= 5 \cdot \int_{3}^{27} \frac{1}{2 u} d u

Take the constant out:

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

= 5 \cdot \frac{1}{2} \cdot \int_{3}^{27} \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 5 \cdot \frac{1}{2} [ln ∣ u ∣]_{3}^{27}

Simplify

5 \cdot \frac{1}{2} [ln ∣ u ∣]_{3}^{27} : \frac{5}{2} [ln ∣ u ∣]_{3}^{27}

= \frac{5}{2} [ln ∣ u ∣]_{3}^{27}

Compute the boundaries:

2 ln (3)

= \frac{5}{2} \cdot 2 ln (3)

Simplify

= 5 ln (3)

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What is the integral from 1 to 5 of (5x)/(x^2+2) ?
The integral from 1 to 5 of (5x)/(x^2+2) is 5ln(3)