### SOLUTIONS TO INTEGRATION BY PARTIAL FRACTIONS

SOLUTION 15 : Integrate . First rewrite the function by multiplying by , getting truein

.

Now use the method of u-substitution. Let

so that

.

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

(Decompose into partial fractions.)

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

(Recall that .)

.

SOLUTION 16 : Integrate . Decompose into partial fractions, getting

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

.

SOLUTION 17 : Integrate . Decompose into partial fractions (There is a repeated linear factor !), getting

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

;

let ;
let
;
it follows that and and ;
let
.)

.

SOLUTION 18 : Integrate . 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 .)

(Recall that .)

.

SOLUTION 19 : Integrate . Use the method of u-substitution first. Let

so that

.

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

(Factor and decompose into partial fractions.)

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

.

SOLUTION 20 : Integrate . Begin by rewriting the denominator by adding , getting

(The factors in the denominator are irreducible quadratic factors since they have no real roots.)

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

;

let
;
it follows that and and ;
let

it follows that and and .)

.

Now use the method of substitution. In the first integral, let

so that

.

In the second integral, let

so that

.

In addition, we can back substitute", using

in the first integral and

in the second integral. Now substitute into the original problems, replacing all forms of , getting

(Recall that .)

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