Calculate $\int_0^1f(x)dx$ if

Question

If $$f(x)=x+\int_0^1{(x+t)tf(t)}dt$$

find

$$\int_0^1f(x)dx$$

Any Hints would be helpful. I have no leads so far.

Answer 1

Hint: Let $$f(x)=mx+c$$ Find $m$ and $c$ first.

Answer 2

Let $\displaystyle A=\int_0^1tf(t)dt$ and $\displaystyle B=\int_0^1t^2f(t)dt$.

Then $f(x)=(A+1)x+B$.

$$A=\int_0^1 tf(t)dt=\int_0^1[(A+1)t^2+Bt]dt=\frac{1}{3}(A+1)+\frac{1}{2}B$$

$$B=\int_0^1 t^2f(t)dt=\int_0^1[(A+1)t^3+Bt^2]dt=\frac{1}{4}(A+1)+\frac{1}{3}B$$

Solve for $A$ and $B$.