limit of integral

[Vali]

I tried to use integration by parts.
I took f(x)=arctan(x) => f'(x)= 1/x^2+1
g'(x)=cos(nx) => g(x)= sin(nx)/n
So I get sin(nx)/n * arctan(x) - integral from 0 to 1 from sin(nx)/n(x^2+1)
How to continue ?
I'm always getting stuck with this kind of exercises ( limits of integrals ) because I don't know how to replace n (infinity) in functions and I noticed that I have to use intervals and inequalities to resolve this kind of limits.
Some ideas?

 

[Dr.Peterson]

I tried to use integration by parts.
I took f(x)=arctan(x) => f'(x)= 1/(x^2+1)
g'(x)=cos(nx) => g(x)= sin(nx)/n
So I get sin(nx)/n * arctan(x) - integral from 0 to 1 from sin(nx)/(n(x^2+1))
How to continue ?
I'm always getting stuck with this kind of exercises ( limits of integrals ) because I don't know how to replace n (infinity) in functions and I noticed that I have to use intervals and inequalities to resolve this kind of limits.
Some ideas?

I suspect that, rather than parts, a substitution may work better; I'd try u = arctan(x).

But it is also possible that the indefinite integral would be hard or impossible, and the problem relies on its being a limit of a definite integral.

What other sorts of limit-of-integral problems have you seen, and what techniques have been used?

 

[Otis]

:idea: Symbol n does not represent infinity because infinity is not a number. :cool: