I am in the middle of a problem and having trouble integrating the following integral:...

Ikunupe6v

Ikunupe6v

Answered

2022-01-05

I am in the middle of a problem and having trouble integrating the following integral:
111(1+x2)2dx

Answer & Explanation

puhnut1m

puhnut1m

Expert

2022-01-06Added 33 answers

Try making a substitution x=tanu. Notice then that
(1+x2)2=(1+tan2u)2=(sec2u)2=(sec2u)2=sec4u
and
dx=sec2udu
So the indefinite integral is now
1sec2udu=cos2udu
This new integrand should be easier to integrate. Just remember to change your limits to get the proper evaluation.
Robert Pina

Robert Pina

Expert

2022-01-07Added 42 answers

If we add and subtract x2 in the numerator, we can integrate the first integral immediately
1(1+x2)2dx=11+x2dxx2(1+x2)2dx
arctanxxx(1+x2)2dx
and the second integral by parts:
xx(1+x2)2dx=x(12(1+x2))+12(1+x2)dx
=x2(1+x2}+12arctanx
user_27qwe

user_27qwe

Expert

2022-01-11Added 230 answers

1(1+x2)2=(11+x2x21+x2)2)=(11+x2+x2(11+x2))=(11+x2+(12x1+x2)1211+x2)=12(arctanx+x1+x2)

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