Solve for $f(x)$: $f(x) = x^2+\int_0^1xf\left(t\right)dt$

Question

I am struggling with this problem: $f(x) = x^2+\int_0^1xf\left(t\right)dt$, and I have to solve for $f(x)$.

What I Did:

$f(x) = x^2+x\int_0^1f\left(t\right)dt$

$a=\int_0^1f\left(t\right)dt$

So, $f(x)=x^2+ax$.

I have tried many things from this point, including substituting $f(t)$ with $f(x)$, but none of it seems to work.

I would be grateful if I got assistance.

Answer 1

\begin{align*} \int_{0}^{1}f(x)dx&=\int_{0}^{1}x^{2}dx+a\int_{0}^{1}xdx\\ a&=\dfrac{1}{3}+\dfrac{1}{2}a\\ a&=\dfrac{2}{3}, \end{align*} so $f(x)=x^{2}+\dfrac{2}{3}x$.