Struggling with Integration by Parts? Try a New Approach with Secant Functions!

[Ginny Mac]

Integration by parts...

I just started Calc. II and though I struggle a bit, it's fascinating. I have been fooling with a problem lately...one of those standard problems that professors like to assign, and it usually appears in calculus texts:

Have ya'll ever done integration by parts with secx? A friend of mine worked it out for me, but I have had trouble reaching the solution on my own. Just thought I'd throw it out there in case ya'll hadn't run into it! It is a fun kind of headache.

-Gin

 

[quantumdude]

Hi Ginny,

I'm scooting this over to our Homework section.

Now, on to your question: Are you talking about integrands that contain powers of $\sec(x)$, or just the $\sec(x)$ itself? The reason I'm asking is that it is normal to integrate odd powers greater than 1 of the secant function by parts, but not so normal to integrate the secant function itself by parts.

 

[Jameson]

That integral doesn't really require integration by parts... You have to mulitply by a form of one, which can be hard to see.

$$\int \sec{x}dx= \int \sec(x)*\frac{\sec(x)+\tan(x)}{\sec(x)+\tan(x)}dx$$

You can do a u-substitution from there.

 

[TD]

Or use the substitution $t = \tan \left( {x/2} \right)$

 

[Benny]

Or $$\sec x = \frac{1}{{\cos x}}\left( {\frac{{\cos x}}{{\cos x}}} \right) = \frac{{\cos x}}{{\cos ^2 x}} = \frac{{\cos x}}{{1 - \sin ^2 x}}$$ and use substitution + partial fractions. It's one of the longer ways of doing it but it is an alternative.