Exponential Function

 

The area under the curve is:

 

To obtain this answer use the substitution u = -ax and du = -adx. Therefore dx = -du / a.

The integral can be written:

 

 In general exponential integrals multiplied by a polynomial can be solved using integration by parts. For example, the integral

can be solved by saying that u = x and dv = e-ax dx. We can use

du = dx and v = (-1/a)e-ax to obtain the integral. In general

so that in the present case we have

 

 

Notice that the first (uv) term is zero and the second term contains the integral of e-ax that was solved above. By this technique any polynomial can be solved yielding the general formula