*u* = *x*^{2}+4*x*-3

so that

*du* = (2*x*+4) *dx* = 2 (*x*+2) *dx* ,

or

(1/2) *du* = (*x*+2) *dx* .

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

.

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* SOLUTION 9 :* Integrate
. Let

*u* = *x*^{2}+1

so that

*du* = 2*x dx* ,

or

(1/2) *du* = *x dx* .

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

(Recall that .)

.

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* SOLUTION 10 :* Integrate
.
Let

so that

.

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

.

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* SOLUTION 11 :* Integrate
.
Let

so that (Don't forget to use the chain rule.)

,

or

.

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

(Do not make the following VERY COMMON MISTAKE : . Why is this INCORRECT ?)

.

Now make another substitution. Let

*w* = -*u*

so that

*dw* = (-1) *du* ,

or

(-1) *dw* = *du* .

Substitute into the problem, replacing all forms of *u*, getting

.

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* SOLUTION 12 :* Integrate
.
Let

*u* = *x*^{2}

so that

*du* = 2*x dx* ,

or

(1/2) *du* = *x dx* .

In addition, the range of *x*-values is

,

so that the range of *u*-values is

,

or

.

Substitute into the original problem, replacing all forms of *x* and the *x*-values, getting

= (-1/2)( (-1) - (1) )

= (-1/2)( -2)

= 1 .

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* SOLUTION 13 :* Integrate
. Let

*u* = *x*-1

so that

*du* = (1) *dx* = *dx* .

In addition, we can "back substitute" with

*x* = *u*+1 .

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

.

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