What is the integral of (tarctan(t))/(sqrt(1+t^2)) ?

The integral of (tarctan(t))/(sqrt(1+t^2)) is arctan(t)sqrt(1+t^2)-ln|sqrt(t^2+1)+t|+C

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integral of (tarctan(t))/(sqrt(1+t^2))

\int \frac{t arctan ( t )}{1 + t ^{2}} d t

arctan (t) 1 + t^{2} - ln t^{2} + 1 + t + C

Solution steps

\int \frac{t arctan ( t )}{1 + t ^{2}} d t

\frac{t arctan ( t )}{1 + t ^{2}} = t arctan (t) \frac{1}{1 + t ^{2}}

= \int t arctan (t) \frac{1}{1 + t ^{2}} d t

Apply Integration By Parts

= arctan (t) 1 + t^{2} - \int \frac{1}{t ^{2} + 1} d t

\int \frac{1}{t ^{2} + 1} d t = ln t^{2} + 1 + t

= arctan (t) 1 + t^{2} - ln t^{2} + 1 + t

Add a constant to the solution

= arctan (t) 1 + t^{2} - ln t^{2} + 1 + t + C

What is the integral of (tarctan(t))/(sqrt(1+t^2)) ?
The integral of (tarctan(t))/(sqrt(1+t^2)) is arctan(t)sqrt(1+t^2)-ln|sqrt(t^2+1)+t|+C