One of the toughest things to do in calculus is to recognize or recall what to do with a given problem in order to solve it. Professors tell you about the problem their trying to teach but they don’t tell you about the subtle differences, what to do if this happens or is changed.I was programming – integrating e^(x) problems on the Titanium or TI-92+, or Voyage 200, for you folks using integration by parts and was really proud of myself – the step by step answers for most problems were coming up using the formula u*v – ∫(v*du) — all of sudden there were wrong answers compared with what the calculator was coming up with using its integral program or system. Why?? So here’s what I find, and have written another program to handle it. Remember I don’t program to learn or teach calculus – I program to pass the pathetic tests, to pass the class, and get hell out of there, never to touch the subject again. So:Anything with e^(x) or e^(ax) (“a” being some arbitrary number) even when connected to sin(x) or tan(x) or ln(x) etc is an integration by parts problem.
Integrating anything with e^(x^2), e^(3x^2), e^(3x-1), e^(5x+6) is U- substitution.
To make it clear — if there is an exponent within the exponent of e, or if there is a minus sign or plus sign within the exponent of e you are using U- substitution.
Now isn’t that fun!! What professor told you about that in class. When I program this stuff – I (and the program) have to know those differences, the program has to work line by line, step by step with the best system possible to solve the problem logically and correctly. Enjoy my programs!