1. Use parts to derive the reduction formula for powers of cosine: for any positive integer n, cos"+(x)ds = cos"(x) sin(x) + cos"-1(x)dx (Hint: cos"+(x) = cos"(r) sin'(x)dx. Also remember that sin' (x) = 1 - cos' (I).) 2. Use parts to find a reduction formula for In(r)"dr. That is, find a formula reducing [ In(x)"dr to an expression involving the antiderivative of some lower power of In(x). (Hint: use du = da.) 3 (Extra Credit). Suppose that f and g have continuous second derivatives, and that both vanish at the endpoints of [a, b) - that is, f(a) = g(a) = f(b) = g(b) = 0 and have continuous second derivatives. Use integration by parts twice to show that 4. (20 points) Use the method of substitution to find the following anti- derivatives: (a) 2xv1 - x'dx (b) tan(x/a)dx (c) redr (d) "in() cos(x)dx (e) 1 r In(r) 5. Given that / f(x)de = 10, find (a) [ f(sin(mz)) cos(Tz)de (b) f(23 - 3)adr