Dfrac11- e -sin x dx is

We know that : nbsp; ∫_b^af(x)dx+∫_b^ag(x)dx=∫_b^a(f(x)+g(x))dx ∴ I=∫_ π/2^π/2(1/1+e^sin x + 1/1+e^ sin x) dx =∫_ π/2^π/2(1/1+e^sin x + e^sin x/1+e^sin x) ...

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Byju's Answer

Standard XII

Mathematics

Property 3

dfrac11+ e ∧-...

Question

The integral value of $\frac{π}{2} \int \frac{- π}{2} \frac{1}{1 + e^{sin x}} d x + \frac{π}{2} \int \frac{- π}{2} \frac{1}{1 + e^{- sin x}} d x$ is ______

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Solution

We know that :
$a \int b f (x) d x + a \int b g (x) d x = a \int b (f (x) + g (x)) d x$

$∴ I = \frac{π}{2} \int \frac{- π}{2} (\frac{1}{1 + e^{sin x}} + \frac{1}{1 + e^{- sin x}}) d x$
$= \frac{π}{2} \int \frac{- π}{2} (\frac{1}{1 + e^{sin x}} + \frac{e^{sin x}}{1 + e^{sin x}}) d x$
$= \frac{π}{2} \int \frac{- π}{2} (\frac{1 + e^{sin x}}{1 + e^{sin x}}) d x$
$= \frac{π}{2} \int \frac{- π}{2} d x$
$= [x]_{\frac{- π}{2}}^{\frac{π}{2}}$
$= \frac{π}{2} - \frac{(- π)}{2} = π$

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The integral value of π2∫−π211+esinxdx+π2∫−π211+e−sinxdx is ______

We know that : a∫bf(x)dx+a∫bg(x)dx=a∫b(f(x)+g(x))dx ∴I=π2∫−π2(11+esinx+11+e−sinx)dx =π2∫−π2(11+esinx+esinx1+esinx)dx =π2∫−π2(1+esinx1+esinx)dx =π2∫−π2dx =[x]π2−π2 =π2−(−π)2=π

The integral value of $\frac{π}{2} \int \frac{- π}{2} \frac{1}{1 + e^{sin x}} d x + \frac{π}{2} \int \frac{- π}{2} \frac{1}{1 + e^{- sin x}} d x$ is ______