evaluate the definite integral below. ∫₁₆¹⁹ 4/x dx

Explanation:

Step1: Recall integral formula

The antiderivative of $\frac{4}{x}$ is $4\ln|x|$ since the integral of $\frac{1}{x}$ is $\ln|x|$ and by the constant - multiple rule of integration $\int kf(x)dx=k\int f(x)dx$ ($k = 4$ here).

Step2: Apply the fundamental theorem of calculus

The fundamental theorem of calculus states that $\int_{a}^{b}F^\prime(x)dx=F(b)-F(a)$. Here $F(x) = 4\ln|x|$, $a = 16$, and $b = 19$. So we have $4\ln(19)-4\ln(16)$.

Step3: Use the property of logarithms

Using the property of logarithms $\ln m-\ln n=\ln\frac{m}{n}$, we can rewrite $4\ln(19)-4\ln(16)$ as $4(\ln(19)-\ln(16))=4\ln\frac{19}{16}$.

Answer:

$4\ln\frac{19}{16}$