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### SOLUTIONS TO INTEGRATION USING A POWER SUBSTITUTION

* SOLUTION 11 :* 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

.

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* SOLUTION 12 :* Integrate
. Use the power substitution

so that

and

.

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

(Use polynomial division.)

.

Use the method of partial fractions. Factor and decompose into partial fractions, getting

(After getting a common denominator, adding fractions, and equating numerators, it follows that
;

let
;

let
.)

(Recall that
.)

.

Click HERE to return to the list of problems.

* SOLUTION 13 :* Integrate
. Use the power substitution

so that

,

,

and

.

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

.

Use the method of partial fractions. Factor and decompose into partial fractions, getting (There are repeated linear factors!)

(After getting a common denominator, adding fractions, and equating numerators, it follows that

;

let
;

let
;

let

;

let

;

it follows that
and
.)

(Recall that
.)

.

Click HERE to return to the list of problems.

* SOLUTION 14 :* Integrate
. Use the power substitution

so that

and

.

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

(Use polynomial division.)

.

Use the method of partial fractions. Factor and decompose into partial fractions, getting

(After getting a common denominator, adding fractions, and equating numerators, it follows that

;

let
;

let
;

let

;

it follows that
and
and .)

.

Click HERE to return to the list of problems.

Duane Kouba
2000-05-09