Hello, Freyr!
We have: .
Let
Substitute: . Got it?
We had a quiz today and there was a problem where I didn't even know where to start... I was hoping someone could help me out on it.
I'm not very good with the syntax to this editor, but I'll try my best.
Evaluate the following integral:
∫ x^4 / (x^10 + 16) dx
Clarification:
Numerator: x^4
Denominator: x^10 + 16
(My problem was I didn't know if I should try substitution, because it doesn't look like I can factor the denominator.)
Thanks,
-Freyr.
NEVER use when the denominator contains a quadratic expression. Use only if it's LINEAR. You can always check your answer by differentiation.
Recall that an integral of the type can be simplified by letting . So in the end, the integral becomes
And note that, in this case, . Either way, you're substituting twice, so do not use the variable . Make sure your final answer contains and not or any other variable.
I hope that helps.
P.S. Maths ALWAYS requires you to think 'out of the box' . That's why it's so much fun.
You can't, unless you want a cookbook that rivals the American constitution for complexity.
There are general principles but ultimately the student's ability and experience is what solves the problem. Especially experience.
It also helps if the student has met and can remember the basic standard forms (like the one that appears in this thread). Unfortunately that's rarely the case.
The mistake the OP made is a consequence of misunderstanding what is in the Cook Book Calculus For Dummies.