Evaluate the integral: I = integral ((2x+3) dx)/(x² + 3x
Evaluate the integral: $$ I = \int \frac{(2x+3)\,dx}{x^2 + 3x - 10}. $$
Explanation of Method
Checking whether the numerator equals the derivative of the denominator is a quick diagnostic for a logarithmic integral: here $\tfrac{d}{dx}(x^2+3x-10)=2x+3$ matches the numerator exactly, so the substitution $u=x^2+3x-10$ reduces the integral to the pure logarithmic form $\int \frac{du}{u}$.
- Factor the denominator (optional, for insight).
- Check whether the numerator equals the denominator's derivative.
- Substitute and integrate using the logarithmic antiderivative.
- Substitute back and add the constant of integration.
Step 1 · Factor the denominator and check the numerator
To solve this integral, we use the substitution method. First, we factorize the denominator:
$$ x^2+3x-10 = (x+5)(x-2). $$
Observing the numerator $2x+3$, we check if it can be rewritten in terms of the derivative of the denominator:
$$ \frac{d}{dx}\left(x^2+3x-10\right) = 2x+3. $$
Since the numerator matches the derivative of the denominator, we let:
$$ u = x^2+3x-10. $$
Now, differentiate both sides:
$$ \frac{du}{dx} = 2x+3. $$
Rearranging, we get:
$$ du = (2x+3)\,dx. $$
Step 2 · Integrate
Thus, the integral simplifies to:
$$ I = \int \frac{du}{u}. $$
The integral of $\dfrac{1}{u}$ is:
$$ I = \ln|u| + C. $$
Step 3 · Substitute back
Substituting back $u = x^2+3x-10$, we obtain:
$$ I = \ln\left|x^2+3x-10\right| + C. $$
Summary Table
| Quantity | Value |
|---|---|
| Factored denominator | $(x+5)(x-2)$ |
| Substitution | $u = x^2+3x-10$ |
| Antiderivative | $\ln|u|$ |
Additional Notes
- The domain of $I$ requires $x \neq -5,\ 2$, where the denominator vanishes.
- The factoring of the denominator into $(x+5)(x-2)$ is not strictly needed for this method — it's included for context, but the logarithmic shortcut works directly once the numerator is recognized as the denominator's derivative.
- The result can be checked by differentiating $\ln\left|x^2+3x-10\right|$ with the chain rule and confirming it returns $\dfrac{2x+3}{x^2+3x-10}$.
Result
The solution to the integral is: