# Integration problem

How does one integrate
$$\int\arctan 4t\,dt$$
?

This is out of a section on integration by parts. Maybe that would help but I don't see any "parts." I only see one, the arctan.

Math Is Hard
Staff Emeritus
Gold Member
Let u = arctan4t
dv = dt

Thank you so much. Please tell me if I did this correctly.
Integration by Parts:
$$\int u\,dv = uv - \int v\,du$$

$u = \arctan 4t , \,dv = dt , v = t\,dt , \,du = \frac{4}{16t^2 + 1}$
$$\int\arctan 4t\,dt = t\arctan 4t - \int\frac{4t}{16t^2 + 1}\,dt = t\arctan 4t - \frac{1}{8}\ln(16t^2 + 1) + C$$

Math Is Hard
Staff Emeritus