## 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.

Practice 1: The integral $\int \frac{1}{25-x^{2}} \mathrm{dx}$ matches table entry 37: $\int \frac{1}{a^{2}-x^{2}} d x=\frac{1}{2 a} \ln \left|\frac{a+x}{a-x}\right|+C$

when we put $a=5$, so $\int \frac{1}{25-x^{2}} \mathrm{dx}=\frac{1}{2 \cdot 5} \ln \left|\frac{5+\mathrm{x}}{5-\mathrm{x}}\right|+\mathrm{C}$.

Practice 2: The integral $\int \frac{1}{7-x^{2}} \mathrm{dx}$ matches table entry 37: $\int \frac{1}{a^{2}-x^{2}} d x=\frac{1}{2 a} \ln \left|\frac{a+x}{a-x}\right|+C$

when $\mathrm{a}=\sqrt{7}, \text { so } \int \frac{1}{7-x^{2}} \mathrm{dx}=\frac{1}{2 \sqrt{7}} \ln \left|\frac{\sqrt{7}+\mathrm{x}}{\sqrt{7-x}}\right|+\mathrm{C}$.

Practice 3: $\int \frac{1}{25-9 x^{2}} \mathrm{dx}=\frac{1}{9} \int \frac{1}{(25 / 9)-x^{2}} \mathrm{dx}=\frac{1}{9} \int \frac{1}{(5 / 3)^{2}-x^{2}} \mathrm{dx}$

so use #37 with $a = 5/3$. Then

$\int \frac{1}{25-9 x^{2}} \mathrm{dx}=\frac{1}{9} \int \frac{1}{(5 / 3)^{2}-x^{2}} \mathrm{dx}=\frac{1}{9} \frac{1}{2(5 / 3)} \ln \left|\frac{5 / 3+\mathrm{x}}{5 / 3-\mathrm{x}}\right|+\mathrm{C}$

$=\frac{1}{30} \ln \left|\frac{5 / 3+\mathrm{x}}{5 / 3-\mathrm{x}}\right|+\mathrm{C}$  or $\frac{1}{30} \ln \left|\frac{5+3 \mathrm{x}}{5-3 \mathrm{x}}\right|+\mathrm{C}$

Practice 4: $\int \frac{\cos (x)}{25-\sin ^{2}(x)} \mathrm{dx}$. Put $\mathbf{u}=\sin (x)$. Then $\mathrm{du}=\cos (\mathrm{x}) \mathrm{d} \mathrm{x}$ so

$\int \frac{\cos (x)}{25-\sin ^{2}(x)} \mathrm{dx}=\int \frac{1}{25-\mathrm{u}^{2}} \mathrm{du}=\frac{1}{2 \cdot 5} \ln \left|\frac{5+\mathrm{u}}{5-\mathrm{u}}\right|+\mathrm{C}=\frac{1}{10} \ln \left|\frac{5+\sin (\mathrm{x})}{5-\sin (\mathrm{x})}\right|+\mathrm{C}$

Practice 5: We can evaluate $\int \cos ^{3}(7 x) d x$ by using the recursion formula from the Table with $n = 3$:

$\text { 20. } \int \cos ^{n}(a x) d x=\frac{\cos ^{n-1}(a x) \sin (a x)}{n a}+\frac{n-1}{n} \int \cos ^{n-2}(a x) d x \text {. }$ Then

\begin{aligned} \int \cos ^{3}(7 x) \mathrm{dx}=& \frac{\cos ^{2}(7 \mathrm{x}) \cdot \sin (7 \mathrm{x})}{7 \cdot 3}+\frac{2}{3} \int \cos (7 x) \mathrm{d} x \\ &=\frac{\cos ^{2}(7 x) \cdot \sin (7 x)}{21}+\frac{2}{3} \cdot \frac{1}{7} \cdot \sin (7 x)+\mathrm{C} \end{aligned}