180 likes | 245 Views
Learn how to integrate using u-substitution and change of variables with step-by-step examples and guidelines. Understand pattern recognition in nested derivatives and check your solutions by taking derivatives. Explore even and odd functions in integral calculus.
E N D
Integration by u-Substitution "Millions saw the apple fall, but Newton asked why." -– Bernard Baruch
Objective • To integrate by using u-substitution
In summary… • Pattern recognition: • Look for inside and outside functions in integral • Determine what u and du would be • Take integral • Check by taking the derivative!
Guidelines for making a change of variables • 1. Choose a u = g(x) • 2. Compute du • 3. Rewrite the integral in terms of u • 4. Evaluate the integral in terms of u • 5. Replace u by g(x) • 6. Check your answer by differentiating
Change of variables for definite integrals • Thm: If the function u = g(x) has a continuous derivative on the closed interval [a,b] and f is continuous on the range of g, then
Even and Odd functions • Let f be integrable on the closed interval [-a,a] • If f is an even function, then • If f is an odd function then