Struggling with Trigonometric Substitution and Anti-Derivatives?

[domesticbark]

anti derive sqrt{1 + x^(-2/3)}


So this isn't actually a homework problem... just a problem that's been bugging me. Supposedly it's possible to do this using u substitution, but I'm having quite a bit of trouble... I've tried making U = x^(-2/3) and I realize that's probably not the best thing to do... Either way I've been able to get everything to a u and get du in there, it's just in comes down to a odd multiplication thing. I'm not looking for an answer really, perhaps somehow to be led in the right direction. Or even just show me how to anti derive sqrt {x^2 + x}. That would help a bit since I'm stuck in a similar scenario.

 

[farleyknight]

Pick $x^{-2/3} = \tan(\theta)^2$, and then $\sqrt{1 + x^{-2/3}} = \sqrt{1 + \tan(\theta)^2} = \sqrt{\sec(\theta)^2}$. You'll have to be sure to find $dx = \frac{d}{d\theta} \tan(\theta)^{-3}$ and further work for the substitution there, and later some integration by parts for what shows up afterwards. But I feel that approach should work.

 

[fzero]

The trick to using substitutions is to turn parts of the integrand into things that you already know how to integrate. You might also have to make multiple substitutions before you get something completely manageable. For instance,

$$\int \frac{du}{\sqrt{1+u^2}}$$

can be done by a substitution $$u=\tan v.$$ In your case, you can manipulate your integrand to a form similar to this (though there's another factor of u to some power that might make things a bit harder.) I think that some substitutions of this form are helpful, but haven't worked the whole thing out myself.

 

[Agent M27]

You should try combining the terms under the root, see if that doesn't help a bit, at this point you should be looking for a substitution. Good luck.

Joe

 

[domesticbark]

So what you guys are saying is I can sub other things in for u? If $$\chi$$^-2/3= U, can I really just make U = tan$$\Theta$$? I'm just a little confused since I'm semi new to calculus. And that makes things a little more complicated. how would I get d$$\Theta$$ to replace du? Ahh I am confused. Either way, thank you for the help. I'm sure I'll figure it out eventually.

 

[farleyknight]

http://en.wikipedia.org/wiki/Trigonometric_substitution#Integrals_containing_a2_.2B_x2" is a special kind of u-substitution. You don't have to use an intermediate $x^{-2/3} = u$ unless you feel more comfortable that way. But it does add more steps and may lead to making another mistake.

If trig substitutions are still new to you, perhaps try practicing by using it on a simpler problem involving $1 + x^2$ alone.