Practice question · Multiple choice
Integration by parts turns ∫u dv into uv − ∫v du - trading one integral for another. Why is that progress rather than circular?
Hints
- Apply it to ∫x·eˣ dx with u = x. What does the leftover integral look like?
- Now try it with u = eˣ instead. Is that an improvement?
Show the answer
D. Because u and dv are chosen so the new integral is easier.
Why
With u = x and dv = eˣ dx, differentiating x turns it into 1 and the remaining integral is trivial, that is the progress. The wrong choice shows it is not automatic: u = eˣ raises the polynomial's degree and leaves a harder integral. LIATE encodes which factors improve on differentiation, and exponentials belong at the dv end because they never simplify.
Practise Integration by Parts
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