Integration (antidifferentiation) is generally more difficult than differentiation. There are no sure-fire methods, and many antiderivatives cannot be expressed in terms of elementary functions. However, there are a few important general techniques. One such technique is the Substitution Method, which uses the Chain Rule “in reverse.”

THEOREM 1 The Substitution Method

Before proceeding to the examples, we discuss the procedure for carrying out substitution using differentials. Differentials are symbols such as du or dx that occur in the Leibniz notations du/dx and In our calculations, we shall manipulate them as though they are related by an equation in which the dx “cancels”: For example,

Now when the integrand has the form we can use Eq. (1) to rewrite the entire integral (including the dx term) in terms of u and its differential du: This equation is called the Change of Variables Formula. It transforms an integral in the variable x into a (hopefully simpler) integral in the new variable u.

Change of Variables Formula for Definite Integrals

Calculate the area under the graph of over [1, 3].