Calculus involving Integrals

Question

Prove that $$\int_0^{\sinh(x)} \sqrt{t^2+1}\,dt =x/2+(\cosh⁡(x)+\sinh⁡(x))/2$$

I am lack of trigonometric properties so if you are so kind to provide me some to be able to solve this equation. I would be thankful.

Answer 1

Your conjecture is clearly wrong, since for $x=0$ the LHS vanishes while the RHS becomes $\frac{1}{2}$. Write $t=\sinh u$ so $dt=\cosh u du$ and the integral is $\int_0^x \cosh^2 u du=\frac{1}{2}\int_0^x (1+\cosh 2u)du=\frac{x}{2}+\frac{1}{4}\sinh 2x$.

Answer 2

Hint:

Use the substitution: $$t=\sinh u,\quad \mathrm dt=\cosh u\, \mathrm du.$$ You'll obtain $$\int_0^{\sinh(x)} \sqrt{t^2+1}\,dt =\int_0^x\cosh^2u\,\mathrm du.$$ Then use the linearisation formula: $$\cosh^2 u=\frac{\cos 2u+1}2.$$