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January 28, 2015

January 28, 2015

Posted by **APpreciative student** on Tuesday, November 23, 2010 at 1:48am.

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]

- AP Calculus BC -
**MathMate**, Tuesday, November 23, 2010 at 7:51amIntegration 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).

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