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

let
;

let
.)

(Recall that .)

.

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* 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
.)

.

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* 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
.)

.

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* 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

let
;

let
.)

(Recall that .)

.

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* 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

let
;

let
.)

.

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* 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 .)

.

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* 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
.)

.

Click HERE to return to the list of problems.

* 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
.)

.

Click HERE to return to the list of problems.

Duane Kouba 2000-05-02