Question $5(1$ point$)$

Consider the real integral: ${\displaystyle \underset{0}{\overset{2}{}}}{e}^{cos(x)}*cos(sin(x))dx$

While $it$ looks challenging $to$ evaluate this with standard techniques $(letme$ know $if$

you see a way$),$ complex analysis $is$ here $to$ save the day. Check the two correct

statements.

Select $2$ correct answer$(s)$

This integral comes $up(as$ real part $of$ a complex integral$)$ when $we$ apply the

Mean Value Equality $to$ the function $cos(z)on$ a circle around $0.$

This integral comes $up(as$ real part $of$ a complex integral$)$ when $we$ apply the

Mean Value Equality $to$ the function exp$(z)on$ a circle around $0.$

This integral comes $up(as$ imaginary part $of$ a complex integral$)$ when $we$ apply

the Mean Value Equality $to$ the function $sin(z)on$ a circle around $0.$

This integral comes $up(as$ real part $of$ a complex integral$)$ when $we$ apply the

Mean Value Equality $to$ the function exp$(cos(z))on$ a circle around $0.$

$In$ fact, the Mean Value Equality applied $to$ exp$(z)on$ a circle around $0$ shows

$(by$ comparing real and imaginary parts$)$ that this integral has value $2,$ and that

the integral ${\displaystyle \underset{0}{\overset{2}{}}}{e}^{sin(x)}*sin(sin(x))dx$ has value $0.$

$In$ fact, the Mean Value Equality applied $to$ exp$(z)on$ a circle around $0$ shows

$(by$ comparing real and imaginary parts$)$ that this integral has value $2,$ and that

the integral ${\displaystyle \underset{0}{\overset{2}{}}}{e}^{cos(x)}*sin(sin(x))dx$ has value $0.$