Let a curve y=y(x) solve (12^-1(e^-x))dx = e^2x (schematic; see source), intersecting the …

Question

Let a curve $y=y(x)$ solve $\cos\left(\dfrac12\cos^{-1}(e^{-x})\right)dx = e^{2x}\dots$ (schematic; see source), intersecting the $y$-axis at $y=-1$; the curve meets the $x$-axis at $(\alpha,0)$. Find $e^\alpha$.

Answer: 2

Accepted Answer

Working through the substitution $\cos^{-1}(e^{-x})=	heta$ and integrating, one obtains $2\sqrt{1-e^{-x}} = \sqrt2(y+1)+C$; using the $y$-intercept condition and setting $y=0$ at $x=\alpha$ gives $e^\alpha=2$.

Solutions · 1