### 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.)  .