Definitely Definite

As ${ e }^{ i(sin\theta ) }=cos(sin\theta )+isin(sin\theta )$ ,

We can say-

$\int _{ 0 }^{ 10\pi }{ { e }^{ cos\theta } } cos(sin\theta )d\theta$ $=$ real part of $\int _{ 0 }^{ 10\pi }{ { e }^{ cos\theta } } (cos(sin\theta ) + isin(sin\theta)) d\theta$

$=$ real part of $\int _{ 0 }^{ 10\pi }{ { e }^{ cos\theta } } ({ e }^{isin\theta)} d\theta$

$=$ real part of $\int _{ 0 }^{ 10\pi }{ { e }^{ cos\theta + isin\theta } } d\theta$

$=$ real part of $\int _{ 0 }^{ 10\pi }{\Large{ e }^{ { e }^{ i\theta }} } d\theta$

$=$ real part of $\int _{ 0 }^{ 10\pi }{\Large\left[ 1+{ e }^{ i\theta }+\frac { { e }^{ 2i\theta } }{ 2! } +\frac { { e }^{ 3i\theta } }{ 3! } +\quad .... \right] } d\theta$

$=$ real part of $\int _{ 0 }^{ 10\pi }{\left[ 1+{ (cos\theta +isin\theta ) }+\frac { { (cos2\theta +isin2\theta ) } }{ 2! } +... \right] } d\theta$

$=$ $\int _{ 0 }^{ 10\pi }{\left[ 1+{ cos\theta }+\frac { { cos2\theta } }{ 2! } +\frac { cos3\theta }{ 3! } +... \right] } d\theta$

$=$ ${ \left[ \theta +sin\theta +\frac { sin2\theta }{ 2\cdot 2! } +\frac { sin3\theta }{ 3\cdot 3! } +... \right] }_{ 0 }^{ 10\pi }$

$=$ $10\pi \approx 10(3.14159) = 31.416$