Question

Evaluate \[ \int_{2}^{3} \frac{dx}{x^2 + x} \]

• A. \(\log\!\left(\dfrac{3}{4}\right)\)
• B. \(\log\!\left(\dfrac{3}{2}\right)\)
• C. \(\log\!\left(\dfrac{9}{8}\right)\)
• D. \(\log\!\left(\dfrac{8}{9}\right)\)

Hint:

Always simplify rational functions using partial fractions before integrating.

Answer 1

The Correct Option is C

Step 1: Use partial fractions.
\[ \frac{1}{x^2 + x} = \frac{1}{x(x+1)} = \frac{1}{x} - \frac{1}{x+1} \]
Step 2: Integrate term by term.
\[ \int_{2}^{3} \left( \frac{1}{x} - \frac{1}{x+1} \right) dx \]
Step 3: Evaluate the definite integral.
\[ = \left[ \log x - \log(x+1) \right]_{2}^{3} \] \[ = \left( \log 3 - \log 4 \right) - \left( \log 2 - \log 3 \right) \] \[ = \log\!\left(\frac{9}{8}\right) \]