Question

Question 10 18.01 Practice Questions for Exam 2... What is going on in the solution? I follow everything to do with putting all the y on the left of the equation and all the x on the right of the equation... but then boom they use tan instead of integrating??? What is happening

Answer 1

the integral evalutes to
\[ \int \frac{dx}{1+x^2}= \tan^{-1} x +C\]

See the notes for lecture 5
http://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2006/lecture-notes/lec5.pdf
where they derive
\[ \frac{d}{dx} \tan^{-1} x = \frac{1}{1+x^2}\]
and if we take the integral of both sides, we get the formula they used.