MAT136H1 Midterm: MAT136 Useful Formulas, Hope this one help you to prepare the test， Ace it！

1 ] dt dx in aju 17. dlscca]_ (soc utan ur, dx a[arctan“)-1 rus 21. dx [sinh u] = (cosh u)", 4,tanh u]-(sech? uju, 27, sic Integration. Formulas du = u + C cos u du sin u + C 14. sec u tan u du secu+ C 16. 18. du mm=.arcsina+C "С ton Miffilin Company, Inc.

From Rogawski ET section 4.9, exercise 45. Use the formulas for the derivatives of f(x) = tan(x) and f (x) = sec(x) to evaluate the following indefinite integrals sec2(9x) dx = +C sec(x + 4) tan(x + 4)dx = +C

Let D be a simply connected region in C and let C be a simple closed curve contained in D. Let f(z) be analytic in D. Suppose that z0 is a point enclosed by C. By the usual formulas, show that By using the fact that the line integral of a complex function with an antiderivative is zero and the above, conclude that Now apply Cauchy's formula to conclude that