AP Calculus BC
posted by APpreciative student on .
Hi! Thank you very much for your help
I'm not sure what the answer to this is; how do I solve?
Find antiderivative of
(1/(x^2))[sec(1/x)][tan(1/x)]dx
I did integration by parts and got to
(1/(x^2))[sec(1/x)] + 2*[antiderivative of (1/(x^3))(sec(1/x))dx]

Integration by parts is the same as any other tool. It's just a tool. You can go around in circles with it... unless you know where you're going.
For this particular problem, I propose to use another tool, substitution.
Did you notice there is the factor (1/x²) at the beginning? What would ∫(1/x²)dx give? ∫d(1/x).
So the integral becomes:
I=∫(1/(x^2))[sec(1/x)][tan(1/x)]dx
=∫[sec(1/x)][tan(1/x)]d(1/x)
=∫sec(y)tan(y)dy
= ... +C
Do remember, however, if and when you have to evaluate a definite integral, the limits have to correspond to the integration variable, which in this case is (1/x).