Question:
\(\int \frac{4x+5}{2x+1}dx=\)
\(\int \frac{4x+5}{2x+1}dx=\)
Show Hint
Divide the numerator by the denominator to write the integrand as a constant plus a simple fraction.
Updated On: Nov 12, 2026
- \(2x+\frac{3}{2}\log |2x+1|+c\)
- \(2x-\frac{3}{2}\log |2x+1|+c\)
- \(x+\frac{3}{2}\log |2x+1|+c\)
- \(x-\frac{3}{2}\log |2x+1|+c\)
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The Correct Option is A
Solution and Explanation
Step 1: Understanding the Concept
The degrees of numerator and denominator are equal, so first divide to simplify the fraction.
Step 2: Rewrite
\(\dfrac{4x+5}{2x+1}=\dfrac{2(2x+1)+3}{2x+1}=2+\dfrac{3}{2x+1}\).
Step 3: Integrate
\(\int2\,dx=2x\) and \(\int\dfrac{3}{2x+1}dx=\dfrac{3}{2}\log|2x+1|\).
Step 4: Result
\(\int\dfrac{4x+5}{2x+1}dx=2x+\dfrac{3}{2}\log|2x+1|+c\).
Final Answer:
The integral is 2x plus three halves log of the modulus of (2x + 1), plus c. \[ \boxed{\text{(A) }2x+\dfrac{3}{2}\log |2x+1|+c} \]
The degrees of numerator and denominator are equal, so first divide to simplify the fraction.
Step 2: Rewrite
\(\dfrac{4x+5}{2x+1}=\dfrac{2(2x+1)+3}{2x+1}=2+\dfrac{3}{2x+1}\).
Step 3: Integrate
\(\int2\,dx=2x\) and \(\int\dfrac{3}{2x+1}dx=\dfrac{3}{2}\log|2x+1|\).
Step 4: Result
\(\int\dfrac{4x+5}{2x+1}dx=2x+\dfrac{3}{2}\log|2x+1|+c\).
Final Answer:
The integral is 2x plus three halves log of the modulus of (2x + 1), plus c. \[ \boxed{\text{(A) }2x+\dfrac{3}{2}\log |2x+1|+c} \]
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