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)`:

The shaded region represents the integral we need to find.

Continues 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)`:

The shaded region represents the integral we just found.

Exercise 4

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

The graph of the solution curve we just found, showing that it 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.)