MATA37H3 Lecture 8: Integration by Parts
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Mata37 - lecture 8 - integration by parts. Integration by parts: if u = f (x), v = g(x) are differentiable, then: Inde nite form: udv = uv udu (cid:90) (cid:90) b (cid:12)(cid:12)(cid:12)(cid:12)b a udu a. De nite form: udv = uv (cid:90) (cid:90) b a: suppose f and g are differentiable. Choose u = ln(x), du = x 2dx, du = + c (cid:90) 2: example 2: compute x ln(x)dx. Choose u = ln(x), du = xdx, du = = 2 ln(2) 0 1. 4 (cid:90) t2etdt: example 3: compute (cid:90) (cid:90) Choose u = t2, dv = etdt, du = 2tdt, v = et. Choose u2 = t, dv2 = etdt, du2 = 1 dt, v2 = et. = t2et 2(tet et) + c: example 4: evaluate arctan(x)dx. Choose u = arctan(x), dv = dx, du = 1 + x2 dx, v = x (cid:90) (cid:90) 1.
