Q.Verify: $\int \dfrac{2x+3}{x^2+3x}\,dx = \log|x^2+3x| + C$

Verify by differentiating the right side and checking it equals the integrand.

Let $F(x)=\log|x^2+3x|+C$. With $u=x^2+3x$ and $u'=2x+3$,

This is exactly the integrand — note the numerator $2x+3$ is precisely the derivative of the denominator $x^2+3x$, the hallmark of a $\int \tfrac{u'}{u}\,dx=\log|u|$ form.

Whenever an integrand has the shape $\dfrac{u'(x)}{u(x)}$, its antiderivative is $\log|u(x)|$. Verifying is even simpler: differentiate the claimed answer and check you land back on the integrand.

Spot the pattern

Here $u=x^2+3x$, and $u'=2x+3$ — which is exactly the numerator. So the integrand is $\dfrac{u'}{u}$, and the natural antiderivative is $\log|u|=\log|x^2+3x|$.

Differentiate to confirm

Let $F(x)=\log|x^2+3x|+C$. By the chain rule,

This is the original integrand, and both are defined for $x\neq 0,-3$, so the domains match.

Method: Verifying $\int \frac{f'(x)}{f(x)}\,dx=\log|f(x)|+C$ by differentiation

Use this for "Verify" questions where the proposed answer is a logarithm — and, more generally, to recognise integrands that are a derivative-over-function.

Steps

Step 1: Differentiate the claimed $\log|f(x)|$.

This standard result is the whole engine of the check.

Step 2: Compute $f'(x)$ for the specific $f$.

Identify $f(x)$ (here $f=x^2+3x$) and differentiate it ($f'=2x+3$).

Step 3: Form the ratio and compare with the integrand.

Write $\dfrac{f'(x)}{f(x)}$ and check it matches the given fraction exactly.

Step 4: State the verdict.

If they agree, the antiderivative is verified. The transferable insight: whenever an integrand's numerator is the derivative of its denominator, the integral is $\log$ of the denominator — spotting this pattern is faster than partial fractions or substitution.

Common Mistakes

Mistake 1: Not checking that the numerator is exactly $f'(x)$.

Why it's wrong: the $\log|f|$ rule applies only when the top is precisely the derivative of the bottom; here $\tfrac{d}{dx}(x^2+3x)=2x+3$ matches, but a different numerator would need adjusting. Correct approach: differentiate the denominator and confirm it equals the numerator.

Mistake 2: Forgetting the absolute value in $\log|f(x)|$.

Why it's wrong: $f(x)=x^2+3x$ can be negative, so $\log(x^2+3x)$ is undefined there; the modulus keeps the antiderivative valid on the whole domain. Correct approach: always write $\log|x^2+3x|$.

Mistake 3: Over-complicating with partial fractions.

Why it's wrong: splitting $\tfrac{2x+3}{x^2+3x}$ is unnecessary work when the numerator already equals the denominator's derivative. Correct approach: recognise the $f'/f$ pattern and verify by differentiation directly.