What Is the Correct Approach to Integrate 2*arctan(x) by Parts?

[PCSL]

Homework Statement

problem: $\int$2arctanx dx
2$\int$arctan dx

u=arctanx
du=1/(1+x²)
v=x
dv=dx

xarctanx-$\int$x/(1+x²)

integrate by parts a second time...

u=x
du=dx
v=arctanx
dv=1/1+x²

xarctanx-$\int$arctanx

My final answer I get it 2xarctanx-2xarctanx+2/x²+1 which is obviously wrong. Thanks.

 

[rock.freak667]

Homework Statement

problem: $\int$2arctanx dx
2$\int$arctan dx

u=arctanx
du=1/(1+x²)
v=x
dv=dx

xarctanx-$\int$x/(1+x²)

Don't go integration by parts here, just do a substitution of u=1+x²

 

[PCSL]

Don't go integration by parts here, just do a substitution of u=1+x²

Thanks, I got the answer. This may seem like a dumb question, but how come integration by parts didn't work for this part?

 

[HallsofIvy]

Homework Statement

problem: $\int$2arctanx dx
2$\int$arctan dx

u=arctanx
du=1/(1+x²)
v=x
dv=dx

xarctanx-$\int$x/(1+x²)

integrate by parts a second time...

u=x
du=dx
v=arctanx
dv=1/1+x²

xarctanx-$\int$arctanx

My final answer I get it 2xarctanx-2xarctanx+2/x²+1 which is obviously wrong. Thanks.[/QUOTE]
Because [itex]2/(x^ 1) is the derivative of arctan(x), not the integral.

Your choice of "u" and "dv" are just the results you got from the first integration by parts so you are just reversing the first integration. What you would correctly get is
$$x arctan(x)- x arctan(x)+ \int arctan(x)dx= \int arctan(x)dx$$
exactly what you started with.