JEE Main 2022 · Mathematics · Indefinite and Definite Integrals · hard

Let f be a continuous function such that f(x)=6x^2+x∫0^1 t f(t) dt+3∫0^1 f(t) dt for all real x. Then f(2) equals

Numerical answer

Let

f

be a continuous function such that

f(x)=6x^2+x\displaystyle\int_0^1 t\,f(t)\,dt+3\int_0^1 f(t)\,dt

for all real

x

. Then

f(2)

equals

Show answer and solution

Answer: 21

Let

A=\int_0^1 f(t)\,dt

and

B=\int_0^1 t f(t)\,dt

, which are constants. Then

f(x)=6x^2+Bx+3A

. Compute

A=\int_0^1(6t^2+Bt+3A)\,dt=2+\frac B2+3A

and

B=\int_0^1(6t^3+Bt^2+3At)\,dt=\frac32+\frac B3+\frac{3A}{2}

. The first gives

-2A=2+\frac B2

; the second gives

\frac{2B}{3}=\frac32+\frac{3A}2

. Solving,

A=-1

B=0

, so

f(x)=6x^2-3

and

f(2)=24-3=21

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