.

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 .)

.

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* 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 .)

.

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* 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

.)

.

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* 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 .)

.

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* 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 .)

.

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* 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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Duane Kouba 2000-05-02