Integration: Question 9

Worked solution

Step 1: Apply the (ax+b)^n rule

Using $\displaystyle\int (ax+b)^n\,dx = \frac{(ax+b)^{n+1}}{a(n+1)}+c$ with $a=2$, $n=-2$:

The new power is $n+1 = -1$, and the denominator is $a(n+1) = 2\times(-1) = -2$.

$\int (2x+3)^{-2}\,dx = \frac{(2x+3)^{-1}}{-2} + c = -\frac{1}{2(2x+3)} + c$

Step 2: Recompute independently by differentiating the candidate answer

$\frac{d}{dx}\left(-\frac{1}{2(2x+3)}\right) = \frac{d}{dx}\left(-\frac12(2x+3)^{-1}\right) = -\frac12 \times (-1)(2x+3)^{-2}\times 2 = (2x+3)^{-2}$

This matches the original integrand exactly, confirming the antiderivative in Step 1 is correct.

Why the other options are wrong

• B: has the wrong sign. This comes from forgetting the negative sign that arises when the power $-2$ becomes $-1$.
• C: forgets to divide by $a=2$, only dividing by the new power $n+1=-1$.
• D: multiplies by $a=2$ instead of dividing by it.

Final answer

$\boxed{\int (2x+3)^{-2}\,dx = -\frac{1}{2(2x+3)} + c} \quad \text{(Option A)}$