Find $f(1)$ from equation $\int_{0}^{x}f(t)dt = x+\int_{x}^{1}tf(t)dt$

You are correct. The problem is with the person who made the question.

The functional equation is incorrect.

The correct functional equation for this function must be :

$$\int_{0}^{x}f(t)dt = x+\int_{x}^{1}tf(t)dt + \ln 2 -1$$

There are always issues, if you are differentiating an expression, and there are constants.

Since taking derivative destroys existence of any constant, often there are mistakes in questions, if not formatted correctly.