How do you integrate #int 1/sqrt(9x^2-18x+13) # using trigonometric substitution?
I can give you a solution but it uses a hyperbolic substitution as oppose to a standard trig substitution but it is still very similar.
We begin by re writing the quadratic under the square root in completed square / vertex form.
We can re write the integral as:
At this point we may like to consider the substitution:
It also follows from this substitution that:
Notice it is not a standard trig function but a hyperbolic function
Now putting this into the integral we get:
We can factor
From the hyperbolic identity:
we can replace the expression under the square root with:
Which simplifies to:
Now reverse the substitution for
You can leave it here or if you want to go a bit further, re-write this in terms of the definition of the