In this section, I'll look at some other integration techniques.
Completing the square.
When an integral contains a quadratic expression --- that is, a quadratic with
a middle term
--- you can sometimes
simplify the integrand by completing the
square. This eliminates the middle term of the quadratic; the
resulting integral can then be computed using (e.g.) trig
substitution.
Example. Compute .
Since
and
, I have
Therefore,
Example. Compute .
Since
and
, I have
Therefore,
Next, I need a trig substitution:
Note that in some cases, an integral containing a quadratic with a
middle term can be integrated in other ways. For example, in this
integral I can let :
This integral can be done using partial fractions:
Fractional powers.
When an integral contains fractional powers , you can often simplify the
integrand using a substitution of the form
Take k to be the least common multiple of the denominators of the fractions that occur in the exponent.
Example. Compute .
Since the least common multiple of the denominators 3 and 4 is 12,
I'll use :
Since the top has a higher power than the bottom, I do a long division:
This gives
Hence, my integral is
Example. Compute .
Since the least common multiple of the denominators 2 and 5 is 10,
I'll use :
Example. Compute
.
The idea here is the same as in the last two examples: I can eliminate a square root by putting a square inside.
Example. Compute .
Let , so
. I get
I'll do this integral by parts:
I do the parts computation and put the x's back using (since
):
Copyright 2019 by Bruce Ikenaga