2-4 - Last Time Two Techniques of Integration Last time we talked about two tools for integrating difficult functions Integration by Substitution(Change

2-4 - Last Time Two Techniques of Integration Last time we...

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Last Time: Two Techniques of Integration Last time we talked about two tools for integrating difficult functions: I Integration by Substitution (Change of Variable) Z x = b x = a f ( u ( x )) dx = Z u = u ( b ) u = u ( a ) f ( u ) dx du du = Z u = u ( b ) u = u ( a ) f ( u ) x 0 ( u ) du = Z u = u ( b ) u = u ( a ) f ( u ) u 0 ( x ) du I Integration by Parts Z udv = uv - Z vdu Z b a udv = uv b a - Z b a vdu
Remember the following “tips” we talked about:IIf you see a compound function, think about trying asubstitution. Eg.,Z21e3xdx=Z63eu13du=13Z63euduIIf you’re integrating the product of a polynomial times afunction that anti-differentiates nicely (likeexor sin(x) orcos(x)), try integration by parts. Eg.,Zxexdx=xex-Zexdx=xex-ex+CIIf you have a polynomial times ln(x), ignore the last tip—differentiate the ln(x) and anti-differentiate the polynomial.Eg.,Zx2ln(x)=x33ln(x)-Zx331xdx=13x3ln(x)-1Zx2dx 3
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