Grade 12Integral Calculus

Can you please give me the step wise solution....Thank you

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Rishit chaturvedi

10 Years agoGrade 12

1 Answer

BALAJI ANDALAMALA

10 Years ago

From the properties of definite integrals ,we have

$\int_{a}^{b}f(x)dx = \int_{a}^{b}f(a+b-x)dx$

$Let\,\,I=\int_{-\pi}^{\pi}\frac{sin^2x}{1+a^x}dx$ ….....................(1)
From the above property , we get

$I=\int_{-\pi}^{\pi}\frac{sin^2x}{1+a^{-x}}dx =\int_{-\pi}^{\pi}\frac{a^xsin^2x}{1+a^{x}}dx$ ….........................(2)

Adding (1) and (2) we get

$I+I=2I=\int_{-\pi}^{\pi}\frac{(1+a^x)sin^2x}{1+a^{x}}dx =\int_{-\pi}^{\pi}sin^2x dx$
$=\int_{-\pi}^{\pi}\frac{1-cos2x}{2}dx = \frac{1}{2}\left [ x-\frac{sin2x}{2} \right ]_{-\pi}^{\pi}$
$\Rightarrow 2I =\pi$
$\therefore I =\frac{\pi}{2}$