Question

Let f be a continuous function: a) Find `f(4) if ∫_0^(x^2) f(t) dt = x sin πx` for all `x`. b) Find `f(4) if ∫_0^ f(x)t^2 dt = x sin πx` for all `x`?

Answer 1

a) `f(4)=pi/2`; b) `f(4)=0`

Explanation:

a) Differentiate both sides.

Through the Second Fundamental Theorem of Calculus on the left-hand side and the product and chain rules on the right-hand side, we see that differentiation reveals that:

`f(x^2)*2x=sin(pix)+pixcos(pix)`

Letting `x=2` shows that

`f(4)*4=sin(2pi)+2picos(2pi)`

`f(4)*4=0+2pi*1`

`f(4)=pi/2`

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b) Integrate the interior term.

`int_0^f(x)t^2dt=xsin(pix)`

`[t^3/3]_0^f(x)=xsin(pix)`

Evaluate.

`(f(x))^3/3-0^3/3=xsin(pix)`

`(f(x))^3/3=xsin(pix)`

`(f(x))^3=3xsin(pix)`

Let `x=4`.

`(f(4))^3=3(4)sin(4pi)`

`(f(4))^3=12*0`

`f(4)=0`