Let $f\left( x \right) = \frac{{{{\sin }^2}\,\,\pi x}}{{1\,\, + \,\,{\pi ^x}}}$. Then , \[\smallint \,\left[ {f\left( x \right)\,\, + \,\,f\left( { - x} \right)} \right]\,\,dx\] is equal to

$\frac{x}{2}\,\, - \,\,\frac{{\cos \,\,\pi x}}{{2\pi }}\,\, + \,\,c$

JEE MATHEMATICS MCQ: INDEFINITE INTEGRATION

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Analyzing +4 | -1

Let

f\left( x \right) = \frac{{{{\sin }^2}\,\,\pi x}}{{1\,\, + \,\,{\pi ^x}}}

. Then ,

\int \,\left[ {f\left( x \right)\,\, + \,\,f\left( { - x} \right)} \right]\,\,dx

is equal to

View answer and explanation

Correct option: D

\frac{x}{2}\,\, - \,\,\frac{{\sin \,\,2\pi \,\,x}}{{4\pi }}\,\, + \,\,c

Option D (

\frac{x}{2} - \frac{{\sin 2\pi x}}{{4\pi }} + c

) is correct because it results from simplifying

f(x) + f(-x)

and then integrating the simplified expression.