7.1 Integration By Parts/11

$\int {\text{arctan(4t)}}dt$

$u={\text{arctan(4t)}}\qquad dv=dt$

$du={\frac {4}{1+(4t)^{2}}}dt\qquad v=t$

$\int {\text{arctan(4t)}}dt=t\cdot {\text{arctan(4t)}}-4\int {\frac {t}{1+16t^{2}}}dt=t\cdot {\text{arctan(4t)}}-{\frac {4}{32}}\int {\frac {1}{u}}=t\cdot {\text{arctan(4t)}}-{\frac {1}{8}}\ln(u)=t\cdot {\text{arctan(4t)}}-{\frac {1}{8}}\ln(1+16t^{2})+C$

${\begin{aligned}&u=1+16t^{2}\\[1ex]&du=32tdt\\[1ex]&{\frac {1}{32}}du=tdt\end{aligned}}$