1. In lectures it was shown that for Z h(x) f (t)dt G(x) = 0 we have d G(x) dx = F 0 (h(x))h0 (x) where F (x) = (a) Using the above, find a formula for Rx f (t)dt. 0 d G(x), dx Z where 0 G(x) = f (t)dt. g(x) (b) Hence find a formula for d G(x), dx where Z h(x) f (t)dt. G(x) = g(x) (c) Using your new formula find d G(x), dx where Z x3 G(x) = t2 + 1dt sin(x) 2. Find dy dx (a) y = for the following: (ln(x))x 23x+1 (b) y = (ln(x))sin(x)+cos(x) (c) y = x4 e3x (2x2 + 3x + 1) 5ln(x) (2 + x)4 (d) y = log5 x3 log2 sin(x) 3. Find a General Solution and a Particular Solution to the following: (a) dy y ln(x) = 0, dx x (b) ex dy xy = 0, dx y(e) = 1 y(0) = ln(5) 4. Find a reduction formula for Z In = Hint: Use the fact that y= y . y 2 xn dx. ax + b