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

The integral from 1 to 3 of ln(x^2+3) is ln(432)+pi/(sqrt(3))-4

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integral from 1 to 3 of ln(x^2+3)

\int_{1}^{3} ln (x^{2} + 3) d x

ln (432) + \frac{π}{3} - 4

Decimal

3.88222 \dots

Solution steps

\int_{1}^{3} ln (x^{2} + 3) d x

Apply Integration By Parts

: [x ln (x^{2} + 3) - \int \frac{2 x ^{2}}{x ^{2} + 3} d x]_{1}^{3}

= [x ln (x^{2} + 3) - \int \frac{2 x ^{2}}{x ^{2} + 3} d x]_{1}^{3}

\int \frac{2 x ^{2}}{x ^{2} + 3} d x = - 23 arctan (\frac{x}{3}) + 2 x

= [x ln (x^{2} + 3) - (- 23 arctan (\frac{x}{3}) + 2 x)]_{1}^{3}

Simplify

= [x ln (x^{2} + 3) + 23 arctan (\frac{x}{3}) - 2 x]_{1}^{3}

Compute the boundaries:

3 ln (12) - 2 ln (2) + \frac{π}{3} - 4

= 3 ln (12) - 2 ln (2) + \frac{π}{3} - 4

Simplify

= ln (432) + \frac{π}{3} - 4

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What is the integral from 1 to 3 of ln(x^2+3) ?
The integral from 1 to 3 of ln(x^2+3) is ln(432)+pi/(sqrt(3))-4