What is the integral from 0 to 2 of ln(x^2+4) ?

The integral from 0 to 2 of ln(x^2+4) is 6ln(2)+pi-4

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integral from 0 to 2 of ln(x^2+4)

\int_{0}^{2} ln (x^{2} + 4) d x

6 ln (2) + π - 4

Decimal

3.30047 \dots

Solution steps

\int_{0}^{2} ln (x^{2} + 4) d x

Apply Integration By Parts

= [x ln (x^{2} + 4) - \int \frac{2 x ^{2}}{x ^{2} + 4} d x]_{0}^{2}

\int \frac{2 x ^{2}}{x ^{2} + 4} d x = 4 (- arctan (\frac{x}{2}) + \frac{x}{2})

= [x ln (x^{2} + 4) - 4 (- arctan (\frac{x}{2}) + \frac{x}{2})]_{0}^{2}

Compute the boundaries:

6 ln (2) + π - 4

= 6 ln (2) + π - 4

What is the integral from 0 to 2 of ln(x^2+4) ?
The integral from 0 to 2 of ln(x^2+4) is 6ln(2)+pi-4