### 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

.

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 .)

.

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 .)

.

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 .)

.