# 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:

int e^udu=e^u+K

It is remarkable because the integral is the same as the expression we started with. That is, e^u.

### Example 1

int3e^(4x)dx

### Example 2

inte^(x^4)4x^3dx

### Example 3

int_0^1 sec^2x e^(tan x)dx

Here's the curve y=sec^2x e^(tan x):

Continued below

### Example 4

In the theory of lasers, we see

E=a int_0^(I_0)e^(-Tx)dx

where a, I_0 and T are constants. Find E.

## Exercises

Integrate each of the given functions.

### Exercise 1

int x\ e^(-x^2)dx

### Exercise 2

int(4\ dx)/(sec\ x\ e\ ^(sin\ x)

### Exercise 3

int_(-1)^1(dx)/(e^(2-3x))

Here is the curve y=1/e^(2-3x):

### Exercise 4

Find the equation of the curve for which (dy)/(dx)=sqrt(e^(x+3)) if the curve passes through (1, 0).

### Application - Volume of Solid of Revolution

The area bounded by the curve y = e^x, the x-axis and the limits of x = 0 and x = 3 is rotated about the x-axis. Find the volume of the solid formed. (You may wish to remind yourself of the volume of solid of revolution formula.)

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