Question

\(\int \frac{4x+5}{2x+1}dx=\)

• A. \(2x+\frac{3}{2}\log |2x+1|+c\)
• B. \(2x-\frac{3}{2}\log |2x+1|+c\)
• C. \(x+\frac{3}{2}\log |2x+1|+c\)
• D. \(x-\frac{3}{2}\log |2x+1|+c\)

Hint:

Divide the numerator by the denominator to write the integrand as a constant plus a simple fraction.

Answer 1

The Correct Option is A

Step 1: Substitution
Let $t=2x+1$, so $dx=\dfrac{dt}{2}$ and $4x+5=2t+3$.

Step 2: Integrate in t
$\int\dfrac{2t+3}{t}\cdot\dfrac{dt}{2}=\int\left(1+\dfrac{3}{2t}\right)dt=t+\dfrac{3}{2}\log|t|$.

Step 3: Back-substitute
$2x+1+\dfrac{3}{2}\log|2x+1|$; the constant 1 merges into $c$, giving $2x+\dfrac{3}{2}\log|2x+1|+c$.

Final Answer:
Substituting t = 2x + 1 leads to the same result. \[ \boxed{\text{(A) }2x+\dfrac{3}{2}\log |2x+1|+c} \]