The antiderivative of arctan(4t)dt by integration of parts.

Answer 1

| = integration symbol
| u dv = uv - | v du
| arctan(4t) dt

u = arctan(4t)
du = 4/(16t^2 + 1) dt
dv = dt
v = t

It should look like this,
= t arctan(4t) - | 4t/(16t^2 + 1) dt

After you set that up, substitute
w = 16t^2 + 1
dw = 32t dt
1/32 dw = t dt

Then, the integration should look like this,

= t arctan(4t) - 1/8 | 1/w dw

Make sure I have the correct variables, t/dt as opposed to x/dx (except where I substitute w and dw for t/dt)

Post back if you get stuck. This is a fairly easy integration (except for the substitutions).