Question

$\int \sqrt{e^{2x}-1}dx$

• A. $\sqrt{e^{2x}-1}+{\sec }^{-1}{e}^{2x}+C$
• B. $\sqrt{e^{2x}-1}-{\sec }^{-1}{e}^{2x}+C$
• C. $\frac{1}{2}\sqrt{e^{2x}-1}-\log |e^x\sqrt{e^{2x}-1}|+C$
• D. None of these

Answer 1

The correct option is C $\frac{1}{2}\sqrt{e^{2x}-1}-\log |e^x\sqrt{e^{2x}-1}|+C$

$\int \sqrt{e^{2x}-1}dx\Rightarrow \int \sqrt{{\left(e^x\right)}^2-1}dx\Rightarrow \frac{e^x}{2}\sqrt{e^{2x}-1}-\log \left|e^x+\sqrt{e^{2x}-1}\right|+c\frac{1}{2}\left[e^x\sqrt{e^{2x}-1}-\log \left|e^x+\sqrt{e^{2x}-1}\right|\right]+c$