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
. 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
take the derivative of
with respect to
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.