Using Tables to Find Antiderivatives

Read this section to learn how to use tables to find antiderivatives. See the Calculus Reference Facts for the table of integrals. Work through practice problems 1-5.

Using "Recursive" Formulas

A recursive formula in the table is one that gives one antiderivative in terms of another antiderivative. Usually the new antiderivative is simpler than the original one. For example, table entry #19 gives the pattern for the antiderivative of \sin ^{\mathrm{n}}(\operatorname{ax}) in terms of the antiderivative of \sin ^{n-2}(a x). If we start with the integral of \sin ^{5}(x), then we can use the table to get an answer in terms of the integral of \sin ^{3}(x). Using the recursion formula again, we can get the integral of \sin ^{3}(x) in terms of the integral of \sin (x), which is easy to integrate.

Table entry 19: \int \sin ^{n}(a x) d x=\frac{-\sin ^{n-1}(a x) \cdot \cos (a x)}{n a}+\frac{n-1}{n} \int \sin ^{n-2}(a x) d x.

Example 5: Use the given recursive formula to evaluate \int \sin ^{3}(5 x) \mathrm{dx}.

Solution: In this example \mathrm{n}=3 and \mathrm{a}=5. Then

\int \sin ^{3}(5 x) \mathrm{dx} &=\frac{-\sin ^{2}(5 x) \cdot \cos (5 x)}{3 \cdot 5}+\frac{2}{3} \int \sin (5 x) \mathrm{dx} \\
&=\frac{-\sin ^{2}(5 x) \cos (5 x)}{15}-\frac{2}{3} \frac{1}{5} \cos (5 x)+\mathrm{C}

Practice 5: Use the recursion formula in the table to evaluate \int \cos ^{3}(7 x) \mathrm{d} x.