Problem 35 - Essential Calculus

Question

Evaluate the definite integral. Use a graphing utility to verify your result. $$ \int_{0}^{4} \frac{5}{3 x+1} d x $$

Step-by-Step Solution

Answer

The solution to the definite integral is \(5 ln 13\).

Step 1: Identify the Integral Form

The given function \(\frac{5}{3x+1}\) is a simple rational function, which can be rewritten as \(5 \int_{0}^{4} \frac{1}{3x+1} dx\). The result of the integral of the function in the form of \(\int \frac{1}{x} dx\) is \(ln|x|\) .

Step 2: Evaluate the Integrals

We can now integrate \(5 \int_{0}^{4} \frac{1}{3x+1} dx\). Apply the properties of integrals and you get \(5 [ ln|3x + 1| ]_{0}^{4}\).

Step 3: Substitute the Limits

Now plug in the limits 0 and 4 into \(5 [ ln|3x + 1| ]_{0}^{4}\), it results in \(5 [ln|3*4 + 1| - ln|3*0 + 1|] = 5 [ln|13| - ln|1|] = 5 [ln 13 - 0] = 5 ln 13\).