### SOLUTIONS TO INTEGRATION OF EXPONENTIAL FUNCTIONS

SOLUTION 6 : Integrate . Use u-substitution. Let

u = 1+2ex

so that

du = 2ex dx ,

or

(1/2) du = ex dx .

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

.

SOLUTION 7 : Integrate . First, square the exponential function, recalling that (A-B)2 = A2 - 2AB + B2 . The result is

(Recall that .)

(Use the properties of integrals.)

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

.

SOLUTION 8 : Integrate . Use u-substitution. Let

u = 1+ex

so that

du = ex dx .

In addition, we can "back substitute" with

ex = u-1 .

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

.

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.)

.