What is the integral of ln(y^2+2) ?

The integral of ln(y^2+2) is yln(y^2+2)+2sqrt(2)arctan(y/(sqrt(2)))-2y+C

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integral of ln(y^2+2)

\int ln (y^{2} + 2) d y

y ln (y^{2} + 2) + 22 arctan (\frac{y}{2}) - 2 y + C

Solution steps

\int ln (y^{2} + 2) d y

Apply Integration By Parts

: y ln (y^{2} + 2) - \int \frac{2 y ^{2}}{y ^{2} + 2} d y

= y ln (y^{2} + 2) - \int \frac{2 y ^{2}}{y ^{2} + 2} d y

\int \frac{2 y ^{2}}{y ^{2} + 2} d y = - 22 arctan (\frac{y}{2}) + 2 y

= y ln (y^{2} + 2) - (- 22 arctan (\frac{y}{2}) + 2 y)

Simplify

= y ln (y^{2} + 2) + 22 arctan (\frac{y}{2}) - 2 y

Add a constant to the solution

= y ln (y^{2} + 2) + 22 arctan (\frac{y}{2}) - 2 y + C

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What is the integral of ln(y^2+2) ?
The integral of ln(y^2+2) is yln(y^2+2)+2sqrt(2)arctan(y/(sqrt(2)))-2y+C