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

so that

*du* = 2*e*^{x} *dx* ,

or

(1/2) *du* = *e*^{x} *dx* .

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

.

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* SOLUTION 7 :* Integrate
.
First, square the exponential function, recalling that
(*A*-*B*)^{2} = *A*^{2} - 2*AB* + *B*^{2} . The result is

(Recall that .)

(Use the properties of integrals.)

(Use formula 3 from the introduction to this section on integrating exponential functions.)

.

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* SOLUTION 8 :* Integrate
. Use u-substitution. Let

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

so that

*du* = *e*^{x} *dx* .

In addition, we can "back substitute" with

*e*^{x} = *u*-1 .

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

.

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* SOLUTION 9 :*Integrate
. First, split the function into two parts, getting

(Recall that .)

(Use formula 2 from the introduction to this section on integrating exponential functions.)

.

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