Vtyjkyuk

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

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

Hint: Substitute $z=(x+frac34)$ after completing the square.

Completing the square leads to:

$int sqrt(sqrt2x+frac32^3/2)^2+frac238, dx$

We can factor out $frac12^3/2$ to make it simpler and get to:

$frac12^3/2intsqrt(3x+4)^2+23, dx$

Now substitute $4x+3=t$, giving $fracdtdx=4Leftrightarrow frac14dt=dx$

You get to solve $frac14intsqrtt^2+23, dt$. (Dont forget the other factor!)

The standard substitution $t=sqrt23tan(u)$. Solving for $u$ you get $u=arctan(frac1sqrt23t)$ and

$dt=sqrt23sec^2(u)$

We have to solve $intsqrt23sec^2(u)sqrtunderbrace23tan^2(u)+23_=23sec^2(u), du=23intsec(u)^3,du$

This integral can be solved with partial integration.

There might be better approaches.

Hint: Use the so-called Euler Substitution:
$$sqrt2x^2+3x+4=xsqrt2+t$$
then we get
$$x=fract^2-43-2tsqrt2$$
$$dx=frac-2sqrt2t^2+6t-8sqrt2(3-2tsqrt2)^2dt$$
and
$$sqrt2x^2+3x+4=frac-sqrt2t^2+3t-4sqrt23-2tsqrt2$$

Do you know the integral?

$$intsqrtx^2+a^2dx=fracx2sqrtx^2+a^2+fraca^22ln(fracx+sqrtx^2+a^2a)+C$$