Let f:0,+∞→R andFx=∫0xftdt. If Fx2=x21+x, then f4=
54
7
4
2
Explanation for correct option:
Finding the value of f4
Given data,
f:0,+∞→R
Fx=∫0xftdt
Fx2=x21+x
Consider the given data as,
Fx2=∫0x2ftdtx21+x=∫0x2ftdt
Differentiate both the side with respect to x
dx21+xdx=ddx∫0x2ftdt
We know that
ddx∫abftdt=fb×dbdx-fxdadx
Then,
⇒dx21+xdx=ddx∫0x2ftdt⇒2x+3x2=fx2×2x-0⇒fx2=2x+3x22x⇒fx2=1+3x2⇒f22=1+3×22⇒f4=1+3⇒f4=4
Hence, the correct answer is option (C).