MAT136H1 Lecture 5: MAT187H1 - Lecture 5 - 7.2 Integration by Parts & 7.3 Trig Integrals & 7.4 Trig Substitution
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Mat187h1 - lecture 5 - 7. 2 integration by parts & 7. 3 trig integrals & 7. 4 trig substitution f (x) g(x)dx. We define: u=f (x) dv=g (x)dx and solve using: udv=uv vdu t 2 sintdt. Let u=t2 dv=sintdt du=2tdt v= cost t 2 sintdt= t2 cost+2 tcostdt=tsint tcostdt=? (use i . P . again) sintdt=tsint+cost +c t 2 sintdt= t2 cost+2tsint +2cost +c "=(2 t2)cost+2tsin+c " Note 1: repeated differentiation reduces the order of polynomial: t2 -> t1 ->t0. Note 2: if we had defined: u=sint dv=t 2 dt du=costdt v =t 3. 3 t 3 costdt t 2 sintdt=t 3. > increasing tn term, t2 -> t3 ->t4 use liate u=cos3x du= 3sin 3 xdx dv=e3x dx v= 1. 3 e3x e3x sin 3 xdx u=sin3 x du=3cos 3xdx dv=e3 xdx v= 1. 2 e3xcos 3x + e3xcos 3x + 1. 3 e3x sin 3x e3 xcos3 xdx e3 x(cos 3 x+sin 3 x)+c e3x cos3 xdx= 1.
