### SOLUTIONS TO INTEGRATION USING A POWER SUBSTITUTION

SOLUTION 7 : Integrate . Because we want to simultaneously eliminate a square root and a cube root, use the power substitution

so that

,

,

,

and

.

Substitute into the original problem, replacing all forms of , getting

(Use polynomial division. PLEASE INSERT A FACTOR OF 6 WHICH WAS ACCIDENTLY LEFT OUT.)

.

SOLUTION 8 : Integrate . Remove the outside" square root first. Use the power substitution

so that

,

,

,

and (Use the chain rule.)

.

Substitute into the original problem, replacing all forms of , getting

.

SOLUTION 9 : Integrate . Remove the cube root first. Use the power substitution

so that

,

,

,

and (Use the chain rule.)

.

Substitute into the original problem, replacing all forms of , getting

.

SOLUTION 10 : Integrate . Remove the outside" square root first. Use the power substitution

so that

,

and (Use the chain rule.)

,

or

.

Substitute into the original problem, replacing all forms of , getting

.