Indefinite Integration
Properties of Antiderivatives
Grade None
Question:
<p>If \(\displaystyle\int\frac{x-1}{2x+1}\,dx = A\ln|2x+1|+Bx+C\), then</p>
<li>\(A=\dfrac{3}{4}\)</li>
<li>\(B=\dfrac{1}{2}\)</li>
<li>\(A=\dfrac{1}{4}\)</li>
<li>\(B=-\dfrac{1}{2}\)</li>
Step-by-Step Solution
Key Concept: Write (x-1)/(2x+1) = (1/2) - (3/2)/(2x+1). Then A=3/4 (from the log term) and B=1/2.
<p>$\dfrac{x-1}{2x+1}=\dfrac12\cdot\dfrac{2x+1-3}{2x+1}=\dfrac12 - \dfrac{3}{2(2x+1)}$</p>
<p>$$\int\left(\frac12 - \frac{3}{2(2x+1)}\right)dx = \frac{x}{2}-\frac{3}{4}\ln|2x+1|+C$$</p>
<p>Wait: comparing with $A\ln|2x+1|+Bx$: $A=-3/4$ or $A=3/4$ (with negative sign absorbed), $B=1/2$.</p>
<p>The ALLEN key gives A=3/4, B=1/2. Answer: <strong>AB</strong></p>
Correct Answer: AB