Question

1. $int_{\frac{1}{2}}^{3}(2 - \frac{1}{x})dx$

Explanation:

Step1: Use integral rules

We know that $\int (a - b)dx=\int a dx-\int b dx$. So, $\int_{1/2}^{3}(2-\frac{1}{x})dx=\int_{1/2}^{3}2dx-\int_{1/2}^{3}\frac{1}{x}dx$.

Step2: Integrate each term

The integral of a constant $a$ is $ax + C$. So, $\int_{1/2}^{3}2dx=2x\big|_{1/2}^{3}=2\times3 - 2\times\frac{1}{2}=6 - 1 = 5$. The integral of $\frac{1}{x}$ is $\ln|x|+C$. So, $\int_{1/2}^{3}\frac{1}{x}dx=\ln|x|\big|_{1/2}^{3}=\ln3-\ln\frac{1}{2}=\ln3+\ln2=\ln(3\times2)=\ln6$.

Step3: Calculate the result

$\int_{1/2}^{3}(2 - \frac{1}{x})dx=5-\ln6$.

Answer:

$5-\ln6$