Limits of Integration in Trig Substitution Lets say you must perform a simple integral with trigonometric substitution - we'll choose $\int{\sqrt{1-x^2}dx}$ from $0$ to $17$.  Now we use $x = \sin(t)$ for our substitution.  Here's my problem: the limits of integration for $x$ range from $0$ to $17$, but $\sin$ never achieves a value greater than $1$.  How do we change the limits of integration to accommodate this?  
Note: I am not interested in writing in the limits at the very end, after back-substituting.

If you are substituting $x=\sin t$, then the resultant integral should be $\displaystyle{\int_{0}^{\frac{\pi}{2}}{\cos^2 t dt}}$
