3. Integration: The Exponential Form
by M. Bourne
By reversing the process in obtaining the derivative of the exponential function, we obtain the remarkable result:
It is remarkable because the integral is the same as the expression we started with: eu.
Example 1: 
Example 2: 
Example 3: 
Here is the LiveMath solution to this problem.
Example 4:
In the theory of lasers, we see
where a, I0 and T are constants. Find E.
Exercises
Integrate each of the given functions.
1. 
Put u = -x2 then du = -2x dx
2. 
We can write the question as:
Put u = sin x then du = cos x dx
3. 
Since -(2 - 3x) = 3x - 2, we can write the question as:
Put u = 3x - 2 then du = 3 dx
So
4. Find the equation of the curve for which 
if the curve passes through (1,0).
We need to find
Put u = x + 3 then du = dx
Now, the curve passes through (1,0)
This means when x = 1, y = 0
So
So K = -2e2
So the required equation is:
