Question

evaluate (be sure to check by differentiating!)
\\(\int \frac{5}{2 + 5x} \\, dx, x \
eq -\frac{2}{5}\\)

\\(\int \frac{5}{2 + 5x} \\, dx = \square\\)
(type an exact answer. use parentheses to clearly denote the argument of each function )

Explanation:

Step1: Choose substitution

Let \( u = 2 + 5x \), then \( du = 5dx \).

Step2: Rewrite integral

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

Step3: Integrate

Integral of \( \frac{1}{u} \) is \( \ln|u| + C \).

Step4: Substitute back

Replace \( u \) with \( 2 + 5x \): \( \ln|2 + 5x| + C \).

Step5: Check by differentiation

Differentiate \( \ln|2 + 5x| + C \): \( \frac{5}{2 + 5x} \), which matches the integrand.

Answer:

\( \ln|2 + 5x| + C \)