Question

Find the integral of: 1 / sqrt(4 + x + x^2) dx with limits of range increasing from e to e^2

Answer 1

${\int_e}^{e^2}\frac{1}{\sqrt{({x+\frac{1}{2}})^2+(\frac{\sqrt{15}}{2})^2}}dx\\ \text{Using the standard integral of tan, we have}\\ \frac{2}{\sqrt{15}}tan^{-1}(\frac{2x+1}{\sqrt{15}})|_e^{e^2}\\ \frac{2}{\sqrt{15}}[tan^{-1}(\frac{2e^2+1}{\sqrt{15}})-tan^{-1}(\frac{2e+1}{\sqrt{15}})]\\$