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Question: If \(\int_{}^{}{ƒ(x)}\)dx = ƒ(x), then \(\int_{}^{}{\{ ƒ(x)\}^{2}}\)dx is equal to –...

If ∫ƒ(x)\int_{}^{}{ƒ(x)}dx = ƒ(x), then ∫{ƒ(x)}2\int_{}^{}{\{ ƒ(x)\}^{2}}dx is equal to –

A

12\frac{1}{2}{ƒ(x)}2

B

{ƒ(x)}3

C

{ƒ(x)}33\frac{\{ ƒ(x)\}^{3}}{3}

D

{ƒ(x)}2

Answer

12\frac{1}{2}{ƒ(x)}2

Explanation

Solution

We have, ∫ƒ(x)\int_{}^{}{ƒ(x)}dx = ƒ(x)

Ž (ƒ(x)) = ƒ(x)

Ž 1ƒ(x)\frac{1}{ƒ(x)}d (ƒ(x)) = dx

Ž log (ƒ(x)) = x + log c

Ž ƒ(x) = cex

Ž {ƒ(x)}2 = c2e2x

Ž ∫{f(x)}2\int \{ f ( \mathrm { x } ) \} ^ { 2 }dx = ∫c2e2x\int_{}^{}{c^{2}e^{2x}}dx = c2e2x2\frac{c^{2}e^{2x}}{2} = 12\frac{1}{2} {ƒ(x)}2.

Hence (1) is the correct answer.