Answer
$$\frac{1}{2}\left[\operatorname{coth}^{-1}(4)-\operatorname{coth}^{-1}(20)\right]$$
Work Step by Step
\begin{aligned} \int_{2}^{10} \frac{1}{4 x^{2}-1} d x &=\int_{2}^{10} \frac{1}{-\left(1-4 x^{2}\right)} d x \\ &=-\frac{1}{2} \int_{2}^{10} \frac{2}{\left(1-(2 x)^{2}\right)} d x \\ &=-\left.\frac{1}{2} \operatorname{coth}^{-1}(2 x)\right|_{2} ^{10} \\ &=\frac{1}{2}\left[\operatorname{coth}^{-1}(4)-\operatorname{coth}^{-1}(20)\right] \end{aligned}
