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

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

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integral of ln(2+x^2)

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

\int_{0}^{8} \frac{1}{64 + t ^{2}} d t

\frac{d}{d x} (1 + 4 x)

f (x) = (\frac{x + 1}{x - 1})^{2}

y = x^{4} + 6 x^{2} - x

\int (4 x^{3} + x^{2} + 5) d x

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