If $\int_x^y f(t) dt$ is independent of $x$ and $f(2) = 2$, if the value of $\int_2^x f(t) dt = k \ln x$ then $k$ is ____.
Step-by-Step Solution
Key Concept: Differentiate the integral equation with respect to $x$ to extract the functional form of $f(x)$.
Given $\int_x^y f(t)dt$, differentiate with respect to $x$ treating $y$ as constant to get $yf(xy) - f(x) = 0$. Substitute $y = \frac{1}{x}$ to obtain $\frac{1}{x}f(1) - f(x) = 0$, so $f(x) = f(1) \cdot \frac{1}{x}$. Then $\int_1^x f(t)dt = f(1)\ln x$. Using the condition with $y = \frac{1}{2}$ and $x = 2$ gives $f(1) = 4$, hence $\int_1^x f(t)dt = 4\ln x$.
Correct Answer: 4