SOLUTIONS TO INTEGRATION BY PARTIAL FRACTIONS

SOLUTION 1 : Integrate . 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 2 : Integrate . Factor and decompose into partial fractions, getting

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

.

SOLUTION 3 : Integrate . Factor and decompose into partial fractions, getting

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

.

SOLUTION 4 : Integrate . Because the degree of the numerator is not less than the degree of the denominator, we must first do polynomial division. Then 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 5 : Integrate . Because the degree of the numerator is not less than the degree of the denominator, we must first do polynomial division. Then factor and decompose into partial fractions, getting

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

.

SOLUTION 6 : Integrate . Factor and decompose into partial fractions, getting

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

;

let ;
let ;
let .)

(Recall that .)

.

SOLUTION 7 : 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 ;
let .)

.

SOLUTION 8 : Integrate . Because the degree of the numerator is not less than the degree of the denominator, we must first do polynomial division. Then factor and 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 ;
let
;
let
;
it follows that and .)

.