Question

evaluate the following integral. include absolute values only when needed.\\(\int \frac{dx}{5x + 2}\\)\\(\int \frac{dx}{5x + 2} = \square\\)(use parentheses to clearly denote the argument of each function.)

Explanation:

Step1: Use substitution method

Let \( u = 5x + 2 \), then \( du = 5dx \), so \( dx=\frac{du}{5} \).

Step2: Substitute into the integral

The integral becomes \( \int\frac{1}{u}\cdot\frac{du}{5} \).

Step3: Simplify and integrate

We can factor out the constant \( \frac{1}{5} \), getting \( \frac{1}{5}\int\frac{1}{u}du \). The integral of \( \frac{1}{u} \) with respect to \( u \) is \( \ln|u| + C \), so we have \( \frac{1}{5}\ln|u| + C \).

Step4: Substitute back \( u = 5x + 2 \)

We get \( \frac{1}{5}\ln|5x + 2| + C \). Since the problem says to include absolute values only when needed, and if we consider the domain where \( 5x + 2>0 \) (we can also write it without absolute value if we assume the function is defined in a region where \( 5x + 2 \) has a constant sign, but following the instruction to use absolute values when needed, we can present it as \( \frac{1}{5}\ln(5x + 2)+C \) if we know \( 5x + 2>0 \), or with absolute value. However, the standard form for this integral is \( \frac{1}{5}\ln|5x + 2| + C \), but let's check the problem statement again. It says "include absolute values only when needed". If we are integrating and the original function \( \frac{1}{5x + 2} \) is defined for \( 5x+2
eq0 \), the antiderivative is \( \frac{1}{5}\ln|5x + 2| + C \). But sometimes, in cases where we know the expression inside the log is positive (for example, if we are integrating over an interval where \( 5x + 2>0 \)), we can write it without absolute value. But to follow the instruction, we can present it as \( \frac{1}{5}\ln(5x + 2)+C \) (assuming \( 5x + 2>0 \)) or with absolute value.

Answer:

\( \frac{1}{5}\ln|5x + 2| + C \) (or \( \frac{1}{5}\ln(5x + 2)+C \) if \( 5x + 2>0 \) is assumed)