∫ arctan(4t) d t {\displaystyle \int {\text{arctan(4t)}}dt}
u = arctan(4t) d v = d t {\displaystyle u={\text{arctan(4t)}}\qquad dv=dt}
d u = 4 1 + ( 4 t ) 2 d t v = t {\displaystyle du={\frac {4}{1+(4t)^{2}}}dt\qquad v=t}
∫ arctan(4t) d t = t ⋅ arctan(4t) − 4 ∫ t 1 + 16 t 2 d t = t ⋅ arctan(4t) − 4 32 ∫ 1 u = t ⋅ arctan(4t) − 1 8 ln ( u ) = t ⋅ arctan(4t) − 1 8 ln ( 1 + 16 t 2 ) + C {\displaystyle \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}
u = 1 + 16 t 2 d u = 32 t d t 1 32 d u = t d t {\displaystyle {\begin{aligned}&u=1+16t^{2}\\[1ex]&du=32tdt\\[1ex]&{\frac {1}{32}}du=tdt\end{aligned}}}
![{\displaystyle {\begin{aligned}&u=1+16t^{2}\\[1ex]&du=32tdt\\[1ex]&{\frac {1}{32}}du=tdt\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/ce51b65e9c6d3b8c392f670280a4487d8c5469c7)