Question

Answer the question attached

Answer 1

Compute the indefinite integral ∫cos(x)/(1+sin(x)+cos(x)).
 = arctan(tan(x/2)) - (1/2) ln(tan^2(x/2) + 1) + C.
Then compute the boundaries.
= 1/4 * (π - ln(4)) - 0
= 1/4 * (π - ln(4))

Answer 2

integral of

$f(x)=\frac{cos x}{1+Cosx+Sinx}\\\\=\frac{cos^2\frac{x}{2}-sin^2\frac{x}{2}}{2Cos^2\frac{x}{2}+2Sin\frac{x}{2}Cos\frac{x}{2}}\\\\=\frac{(cos\frac{x}{2}+sin\frac{x}{2})(cos\frac{x}{2}-sin\frac{x}{2})}{(cos\frac{x}{2}+sin\frac{x}{2})(2cos\frac{x}{2})}\\\\=\frac{1}{2}-\frac{sin\frac{x}{2}}{cos\frac{x}{2}}*\frac{1}{2}\\\\=\ \textgreater \ \ \int\limits^{\frac{\pi}{2}}_0 {f(x)} \, dx=[\frac{x}{2}+Ln(Cos\frac{x}{2})]_0^\frac{\pi}{2}\\\\\frac{\pi}{4}-\frac{1}{2}Ln_e 2.$

You could also express cos x and sin x  in terms of  tan x/2  and  then  say u = tan x/2.  Express integrand in terms of  u.  and integrate that.