Question

1. if (f(x)=int_{1}^{x}\frac{2}{t}dt), then what is (f(2))? (a) - 1 (b) (\frac{1}{2}) (c) 1 (d) (ln(2)) (e) (ln(4))

Explanation:

Step1: Recall integral formula

The integral $\int\frac{1}{t}dt=\ln|t| + C$.

Step2: Evaluate $f(2)$

Given $f(x)=\int_{1}^{x}\frac{1}{t}dt$. Substitute $x = 2$ into the integral. Then $f(2)=\int_{1}^{2}\frac{1}{t}dt$.
Using the fundamental - theorem of calculus, if $F(t)=\ln|t|$ is an antiderivative of $\frac{1}{t}$, then $\int_{1}^{2}\frac{1}{t}dt=F(2)-F(1)$.
Since $F(2)=\ln(2)$ and $F(1)=\ln(1) = 0$, we have $f(2)=\ln(2)$.

Answer:

d. $\ln(2)$