Question

${ \underline{ \underline{ \bigstar \: \: \: \pmb{ \sf{Evaluate \: \: this \: \: Integral \: : }}}}}$
$\int_{0} ^{2\pi} \frac{1}{1 + {2}^{ \sin(x) } } dx$
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Answer 1

$\large\underline{\sf{Solution-}}$

Given integral is

$\rm :\longmapsto\:\displaystyle\int_{0} ^{2\pi}\sf \dfrac{1}{1 + {2}^{sinx} }dx$

Let assume that

$\rm :\longmapsto\:Let \: I \: = \: \displaystyle\int_{0} ^{2\pi}\sf\: \dfrac{1}{1 + {2}^{sinx} }dx - - (1)$

We know,

$\rm :\longmapsto\:\displaystyle\int_{0} ^{a}\sf \: f(x) \: dx \: = \: \displaystyle\int_{0} ^{a}\sf f(a - x) \: dx$

So,

$\rm :\longmapsto\:\: I \: = \: \displaystyle\int_{0} ^{2\pi}\sf\: \dfrac{1}{1 + {2}^{sin(2\pi - x)} }dx$

$\rm :\longmapsto\:\: I \: = \: \displaystyle\int_{0} ^{2\pi}\sf\: \dfrac{1}{1 + {2}^{ - sinx} }dx$

$\red{\bigg \{ \because \:sin(2\pi - x) = - \: sinx \bigg \}}$

$\rm :\longmapsto\:\: I \: = \: \displaystyle\int_{0} ^{2\pi}\sf\: \dfrac{1}{1 + \dfrac{1}{ {2}^{sinx} } }dx$

$\rm :\longmapsto\:\: I \: = \: \displaystyle\int_{0} ^{2\pi}\sf\: \dfrac{ {2}^{sinx} }{ {2}^{sinx} + 1}dx - - - (2)$

On adding equation (1) and (2), we get

$\rm :\longmapsto\:\:2 I \: = \: \displaystyle\int_{0} ^{2\pi}\sf\: \dfrac{ 1 + {2}^{sinx} }{ {2}^{sinx} + 1}dx$

$\rm :\longmapsto\:\:2 I \: = \: \displaystyle\int_{0} ^{2\pi}\sf\: 1 \: dx$

$\rm :\longmapsto\:\:2 I \: = \: \bigg( x\bigg)_{0} ^{2\pi}$

$\rm :\longmapsto\:\:2 I \: = \: 2\pi - 0$

$\rm :\longmapsto\:\: I \: = \: \pi$

Additional Information :-

$\rm :\longmapsto\:\displaystyle\int_{a} ^{b}\sf \: f(x) \: dx \: = \: \displaystyle\int_{a} ^{b}\sf f(a + b- x) \: dx$

$\rm :\longmapsto\:\displaystyle\int_{a} ^{b}\sf \: f(x) \: dx \: = \: \displaystyle\int_{a} ^{b}\sf f(y) \: dy$

$\rm :\longmapsto\:\displaystyle\int_{a} ^{b}\sf \: f(x) \: dx \: = - \: \displaystyle\int_{b} ^{a}\sf f(x) \: dx$

$\rm :\longmapsto\:\displaystyle\int_{0} ^{2a}\sf \: f(x) \: dx \: = \: 2\displaystyle\int_{0} ^{a}\sf f( x) \: dx \: \: if \: f(2a - x) = f(x)$

$\rm :\longmapsto\:\displaystyle\int_{0} ^{2a}\sf \: f(x) \: dx \: = 0 \: \: if \: f(2a - x) = - f(x)$

$\rm :\longmapsto\:\displaystyle\int_{ - a} ^{a}\sf \: f(x) \: dx \: = \: 2\displaystyle\int_{0} ^{a}\sf f( x) \: dx \: \: if \: f( - x) = f(x)$

$\rm :\longmapsto\:\displaystyle\int_{ - a} ^{a}\sf \: f(x) \: dx \: = 0 \: \: if \: f( - x) = - f(x)$