### SOLUTIONS TO INTEGRATION OF EXPONENTIAL FUNCTIONS

SOLUTION 1 : Integrate . By formula 1 from the introduction to this section on integrating exponential functions and properties of integrals we get that

.

SOLUTION 2 : Integrate . By formula 1 from the introduction to this section on integrating exponential functions and properties of integrals we get that

.

SOLUTION 3 : Integrate . Use u-substitution. Let

u = 7x

so that

du = 7 dx ,

or

(1/7) du = 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, getting

(Recall that .)

= 4 - 2

= 2 .

SOLUTION 4 : Integrate . Use u-substitution. Let

u = 2x+3

so that

du = 2 dx ,

or

(1/2) du = dx .

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

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

(Recall that .)

.

SOLUTION 5 : Integrate . First, multiply the exponential functions together. The result is

(Recall that and .)

(Use the properties of integrals.)

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

.