Integral of $\int\ e^{-x/2}\sqrt(sin-1)/(cosx+1)$

Question

$\int\ e^{-x/2}\sqrt(sin-1)/(cosx+1)$

The result that I'm getting here contains a cosecx term but the answer has secx term! Please help.

Answer 1

Do you know the integral?

$$\int{\sqrt{x^2+a^2}}dx=\frac{x}{2}\sqrt{x^2+a^2}+\frac{a^2}{2}\ln(\frac{{x+\sqrt{x^2+a^2}}}{a})+C$$

Answer 2

Hint: Substitute $z=(x+\frac{3}{4})$ after completing the square.

Answer 3

Hint: Use the so-called Euler Substitution: $$\sqrt{2x^2+3x+4}=x\sqrt{2}+t$$ then we get $$x=\frac{t^2-4}{3-2t\sqrt{2}}$$ $$dx=\frac{-2\sqrt{2}t^2+6t-8\sqrt{2}}{(3-2t\sqrt{2})^2}dt$$ and $$\sqrt{2x^2+3x+4}=\frac{-\sqrt{2}t^2+3t-4\sqrt{2}}{3-2t\sqrt{2}}$$