evaluate the definite integral below. ∫₂⁴ 1/(3x + 2) dx

Explanation:

Step1: Use substitution

Let $u = 3x+2$, then $du=3dx$. When $x = 2$, $u=3\times2 + 2=8$. When $x = 4$, $u=3\times4+2 = 14$. And $dx=\frac{1}{3}du$.

Step2: Rewrite the integral

The integral $\int_{2}^{4}\frac{1}{3x + 2}dx$ becomes $\frac{1}{3}\int_{8}^{14}\frac{1}{u}du$.

Step3: Integrate $\frac{1}{u}$

The antiderivative of $\frac{1}{u}$ is $\ln|u|$. So, $\frac{1}{3}\int_{8}^{14}\frac{1}{u}du=\frac{1}{3}[\ln(u)]_{8}^{14}$.

Step4: Evaluate the definite - integral

$\frac{1}{3}(\ln(14)-\ln(8))=\frac{1}{3}\ln(\frac{14}{8})=\frac{1}{3}\ln(\frac{7}{4})\approx\frac{1}{3}(0.5596)\approx0.19$.

Answer:

$0.19$