- - 2- - 2e - sin x -cos x-1 - etan xdx is equal to

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Standard XII

Mathematics

Property 3

∫-π / 2π / 2e...

Question

$π / 2 \int - π / 2 \frac{e^{| sin x |} cos x}{(1 + e^{tan x})} d x$ is equal to

e + 1

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1 - e

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e - 1

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1 - \frac{1}{e}

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Solution

The correct option is C $e - 1$
Let $I = π / 2 \int - π / 2 \frac{e^{| sin x |} cos x}{(1 + e^{tan x})} d x$
We know that,
$a \int - a f (x) d x = a \int 0 (f (x) + f (- x)) d x$
$= π / 2 \int 0 (\frac{e^{| sin x |} cos x}{1 + e^{tan x}} + \frac{e^{| sin x |} cos x}{1 + e^{- tan x}}) d x$
$= π / 2 \int 0 e^{| sin x |} cos x d x$
$= π / 2 \int 0 e^{sin x} cos x d x$
$= [e^{sin x}]_{0}^{π / 2} = e - 1$

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π/2∫−π/2e|sinx|cosx(1+etanx)dx is equal to

The correct option is C e−1Let I=π/2∫−π/2e|sinx|cosx(1+etanx)dx We know that, a∫−af(x) dx=a∫0(f(x)+f(−x)) dx =π/2∫0(e|sinx|cosx1+etanx+e|sinx|cosx1+e−tanx)dx =π/2∫0e|sinx|cosxdx =π/2∫0esinxcosxdx =[esinx]π/20=e−1

$π / 2 \int - π / 2 \frac{e^{| sin x |} cos x}{(1 + e^{tan x})} d x$ is equal to