7-3 Integration by Parts

Remember: *If we integrate both sides and rearrange it … Integration by Parts Formula “ultra-violet voo-doo” *Smart choosing of u and v will make the second integral easier to evaluate

Wouldn’t it be nice if we knew which way to choose them? Here’s a helpful saying … L-I-A-T-E L = log I = inv trig A = alg T = trig E = exp Whichever comes first in the list … let u = … second in the list … let dv = One special mixed case: e and sin/cos

Ex 1) Evaluate Let u = x v = sin x du = dx dv = cos x dx Why do we have to do integration by parts in a certain order? Because won’t work, but will. *Note: Integration by parts does not always work!

Ex 2) Evaluate (Repeated use of integration by parts) Let u = x2 v = ex du = 2x dx dv = ex dx Let u = 2x v = ex du = 2 dx dv = ex dx (an easier way will be covered later today)

Confirm using slope field Y1 = x ln x Do Slopefield Ex 3) Solve the differential equation subject to the initial condition y = –1 when x = 1. Confirm the solution graphically by showing that it conforms to the slope field. initial (1, –1): Confirm using slope field Y1 = x ln x Do Slopefield Window [0, 5] x [–5, 5] Graph Y2

Ex 4) Evaluate Let u = ex v = sin x du = ex dx dv = cos x dx Let u = ex v = –cos x du = ex dx dv = sin x dx If you have to do it twice, keep choices for u & dv (don’t switch or you’ll undo)

f (x) and its derivatives g (x) and its antiderivatives Tabular Integration Used as a shortcut for integration by parts, if it meets certain criteria * f (x) goes to 0 and * g (x) loops f (x) and its derivatives g (x) and its antiderivatives (+) (–) (+) etc.

Ex 5) Evaluate (+) (–) (+) (same as Ex 2 – compare answers )

Ex 6) Evaluate (+) (–) (+) (–)

homework Pg. 350 #1 – 26 (skip mult of 3)