UPSC NDA Math Model Paper 5

Let f(x) = xsinx, put -x instead of x then we get f(−x)=(−x)‌sin(−x)=x‌sinx .
Therefore, f(-x) = f(x) implies that f(x) is an even function.
Thus, the integral is given by,

π 2

∫

−π 2

x‌sin‌x‌d‌x =

π 2

∫

0

x‌sin‌x‌d‌x
=2[x(−cos‌x)−∫(−cos‌x‌d‌x)]

π

2

=2[−x‌cos‌x+sin‌x]

π

θ

=2[(−

π

2

‌cos(

π

2

)+sin(

π

2

)) −(−0‌cos‌0+sin‌0)]
=2
Hence, the value of the given integral is 2.