try \int_0^a \sqrt{1+t^2}~dt

or

\int_0^a \sqrt{(1+t)^2}~dt

depending on which you were shooting for.

Consider \sqrt{ } to be a function, so anything inside must be within those brackets. The tilde creates a space. the caret will raise the object to it's left to the power of the first digit to it's right, so a^bc = $\displaystyle a^bc$ if you want a^(bc) you need to write it as a^{bc} = $\displaystyle a^{bc}$

Generally if you are unsure, place {} around your terms, when you get more comfortable with it, you can start using shortcuts like \frac 12 = \frac{1}{2}. So you could write 5/(3x+2) as \frac{5}{3x+2} or as \frac 5{3x+2} and you could write (3x+2)/5 as \frac{3x+2}{5} or \frac {3x+2}5