Integration

[kangkan]

prove that integration from 0 to 1> x^4.cos x<0.2 lies between 0 and 0.2..without obviously solving the integral

[Amit Goyal]

Using
0 <= x^4 (cos x) <= x^4  for all x in (0, 1)
If we integrate the respective expressions in above
inequality over the range (0, 1), the inequality will be
maintained. So integral of 0 from 0 to 1 gives us 0.
And integral of x^4 from 0 to 1 gives us 0.2. Thus, the
desired integral lies between 0 and 0.2.

[SINGHAM]

Are, Hann! Thank you so much, Sir!

Though such blinks never come in my mind during exam!