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

The integral of ln(2+t^2) is tln(2+t^2)+2sqrt(2)arctan(t/(sqrt(2)))-2t+C

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integral of ln(2+t^2)

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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