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### 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.)

.

Click HERE to return to the list of problems.

* 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

.

Click HERE to return to the list of problems.

* 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

.

Click HERE to return to the list of problems.

* 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

.

Click HERE to return to the list of problems.

Duane Kouba
2000-05-09