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Introduction to Integrals

Methods of Calculating Integrals

Problems for "Average Value and Second Fundamental Theorem"

Problems for "Methods of Calculating Integrals"

u-Substitution

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

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:

Figure %: Using trapezoids to approximate area with 3 subdivisions

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.

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