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18.01 Section, November 9, 2015 Section leader: Eva Belmont ([email protected], E18-401B) . Integration techniques: parts, substitution; separable ODE’s 1. Evaluate the following integrals: Z 4 2 (a) x ex dx 2 Z (b) x√ t ln t dt (assume x > 1) 1 Z (c) π sin3 x dx 0 2. Instead of actually doing the following integrals (or even picking up a pen), just talk to your neighbor about how you would go about doing them – e.g. “integration by parts with f = sin x and g = 2x”. Z Z Z Z x 1 dx dx t sin 2t dt x ln x dx 2 1+x 1 + x2 Z 1 + x2 dx x Z Z 2 x3 ex dx tan x dx 1 3. Find the general solution for dy dx = y 2 + 1. 4. Make up an integration by substitution problem and give it to your neighbor. Of course, you should actually know how to do the problem before giving it to someone else! Z 5. Bonus problem: ln x dx Hint: integration by parts. Z 6. Bonus problem: √ sin x dx Hint: first make a substitution. Review • Integration by parts: R f g dx = F g − R F g 0 dx where F = 2 R f.