Find, without using a calculator, integral of 1/sqrt(15+2x-x^2) dx, between 3 and 5, giving your answer as a multiple of pi

To get the denominator into something usable, you have to complete the square so you have it in one of the forms you can use a trig or hyperbolic substitution for. The minus sign in front of the x2 means we'll be aiming for the form a2 - u2 where u is some function of x. The coefficient of x is 2, so u = x - 1 is a good thing to try. Completing the square gives you a -1, which with the - in front of u2 gives + 1. Conviently this means the denominator comes to sqrt(16 - (x-1)2) which means a = 4. This is always a good thing to check for as nine times out of ten you'll get a perfect square.

At this point, you can either go straight for the formula booklet or use a substitution. Always do the first one unless the question specifically states using a substitution. In the later case, you can always look at the booklet to see what substitution you need. It tells you the answer is arcsin, so a sin substitution will work. Either way, the indefinite integral will be arcsin(x-1/4). Evaluating at the limits gives the answer as (arcsin(1) - arcsin(1/2)) = (pi/2 - pi/6) = pi/4.