Trig Substitution vs. Regular Substitution for Indefinite Integrals

[sapiental]

Hi

If I have the indefinite integral x(sqrt(4-x^2))dx

must I replace x with 2sint and then dx = 2cosdt

or can I just do regular substitution with u = 4-x^2 since xdx occurs in the integral already.

when I solve it this way I get -1/3(4-x^2)^3/2 + C

it just seems a lot more work to do it the trigonometric substitituon way.

Any help is much appreciated. Thanks!

 

[courtrigrad]

you have to use the substitution x = 2\sin \theta.

Thus dx = 2\cos \theta d\theta.

At the end you have to solve for \theta to convert the integral back in terms of x. you know that \sin \theta = \frac{x}{2}, so use a right triangle to express this relationship.

 

[sapiental]

hey,

are you positive? My professor Just emailed me saying that the substitution u = 4-x^2 is acceptable.

Did you miss my x infront of the sqrt(4-x^2)dx ?

Really confused now..

 

[arildno]

There is nothing wrong with your own substitution.
You might check, though, that the two substitutions yield the same answer.