Sometimes, it is impossible to find the integral of a function using the methods already discussed. The method of u-substitution allows for the integration of more complex functions, and is essentially a form of the chain-rule in reverse.
The general idea is to change a complex integral involving the variable x and dx into a simpler integral involving the variable u and du . The simpler integral can be evaluated, and then the relation between u and x can be used to express the result in terms of x .
Example: Find
(4x - 2)^{3}
dx
This indefinite integral cannot be evaluated directly by the methods already presented. The first step of a u-substitution
is to pick an appropriate part of the original function to be equal to
u
. This step is often a matter of trial and
error. In this case, however, the choice seems obvious. Let
u = 4x - 2
. Now, to find the relationship between
du
and
dx
,
take the derivative of
u
with respect to
x
:
= 4
Rearranging, this means that
du = 4dx
, or
dx =
du
.
With this information, we can substitute
u
and
du
into the original integral:
(4x - 2)^{3} dx = (u)^{3}()du |
To evaluate this integral, the constant can first be pulled out: = u ^{3} du . By basic integration rules, this is equal to u ^{4} . Now, this result should be translated back into a function of the variable x by using the relation u = 4x - 2 . Substituting x back into the formula yields:
(4x - 2)^{4}
Although other problems may require more trial and error to choose the right definition of u , but all such problems will have the same procedure as the one outlined above. See the problem set for more examples.
The trapezoid rule gives a formula with which we can numerically approximate the value of definite integrals. In previous sections, the area under the curve was approximated using rectangles. The trapezoid rule is based on a method of approximating via trapezoids, as shown below:
The formula for the area of a trapezoid is (base)(average height).
Substituting in the variables in the diagram, this becomes
A Δx + Δx + Δx |
Generalizing from the example and consolidating terms generates the trapezoid rule:
f (x)dx f (x _{0})+2f (x _{1})+2f (x _{3})+...+2f (x _{n-1})+f (x _{n}) |
As was the case with the other approximations, n is the number of subdivisions,
Δx = , andx _{k} = a + kΔx. |
Note that as the number of subdivisions increases, the accuracy of the approximation increases.