Find the indefinite integral.
${\int }_{}^{}\frac{1}{2x+5}dx$

• A. $\ln \mid 2x+5\mid +C$
• B. $\frac{1}{2}\ln \mid 2x+5\mid +C$
• C. $2x+5+C$
• D. $\frac{3}{2x^3+15x}+C$

Answer: (B)

Verified step by step guidance

1. Step 1: Recognize that the integral is of the form ∫(1/(ax+b)) dx, which suggests the use of a logarithmic rule for integration. The general formula for this type of integral is ∫(1/(ax+b)) dx = (1/a) * ln|ax+b| + C.
2. Step 2: Identify the values of 'a' and 'b' in the given integral. Here, the denominator is 2x + 5, so a = 2 and b = 5.
3. Step 3: Apply the formula ∫(1/(ax+b)) dx = (1/a) * ln|ax+b| + C. Substitute a = 2 and b = 5 into the formula. This gives (1/2) * ln|2x+5| + C.
4. Step 4: Simplify the expression to ensure clarity. The result of the integration is (1/2) * ln|2x+5| + C, where C is the constant of integration.
5. Step 5: Verify the result by differentiating (1/2) * ln|2x+5|. The derivative should return the original integrand, 1/(2x+5), confirming the correctness of the solution.