How do you integrate #int (2-x^2)/sqrt(x^2-4)dx# using trigonometric substitution?
2 Answers
Explanation:
Let
Now:
and as
then:
Using the linearity if the integral:
The first integral is standard:
Now solve:
integrate by parts:
Putting it together:
and undoing the substitution, as:
Make a substitution that removes the square root in the denominator via the use of a trig identity.
Explanation:
Approach
We seek to make a substitution that removes the square root in the denominator via the use of a trig identity. Letting
Note that we might have an easier time of the later integration if we used hyperbolic functions instead - we could use
They are both interesting and useful, as well as surprisingly easy to deal with. Knowing about them will enrich your knowledge of the trig functions.
Trig substitution and solution
Substitute
(Use the above identity)
The integrals of both
We'll just take the results:
So, for our integral
Substitute back to our original variable
Alternative solution using hyperbolic substitution
As I mentioned this above as an alternative approach, let's work it through to give you more info on ways to attack such problems.
Instead of substituting
This is where we see how the hyperbolic approach produces easier working - this is an easier integral than the
Substitute back to our original variable
Moral
When presented with an integral of a fraction with a square root denominator of a quadratic, it's always worth taking a moment to think about which particular trig or hyperbolic substitution will make for the easiest route to the answer. There's usually more than one option, and some will be harder work than others. If you can avoid using