MATH137 Lecture : FTC 1 & 2, Indefinite Integrals, FTC proofs, Substitution Method

5. The Second Fundamental Theorem of Calculus. Let f be a continuous function on an interval [a, oo). Define the function F on (a, 0o) by Fr)-f(t)dt. Then F is an anti-derivative off. That is, F"(z) = f(x). (a) Let G(z)= / e-2t dt for z 20. what is G'(x). [lint: Use the Second Fundamental Theorem of Calculus along with the chain rule.] (b) Find the exact value of /e -2tdt Hint: Define F(r) 「e-2tdt. Show that F(a) (1), then take the limit of F(a) as r approaches infinity

(FTC 11) Fundamental Theorern of Calculus, Part 11: 1ff is continuous on [a,b], then where F is any antiderivative of f, that is, a function such that Ff 3. Evaluate the following definite integrals (b)Ved