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

The integral of ln(x^2+2) is xln(x^2+2)+2sqrt(2)arctan(x/(sqrt(2)))-2x+C

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integral of ln(x^2+2)

integral

ln (x^{2} + 2)

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

8 x - 3

\frac{( 2 x + 5 )}{( 7 x + 22 )}

tan^{3} (x) sec^{3} (x)

x + 5

(1 - x)^{\frac{1}{2}}

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