# Miscellaneous Substitutions

When an integral contains a quadratic expression , 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:

When an integral contains fractional powers , you can often simplify the integrand using a substitution of the form

where k is 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.