MHT CET · Maths · Indefinite Integration
\(\int_{2}^{3} \frac{d x}{x^{2}+x}=\)
- A \(\log \left(\frac{3}{4}\right)\)
- B \(\log \left(\frac{3}{2}\right)\)
- C \(\log \left(\frac{9}{8}\right)\)
- D \(\log \left(\frac{8}{9}\right)\)
Answer & Solution
Correct Answer
(C) \(\log \left(\frac{9}{8}\right)\)
Step-by-step Solution
Detailed explanation
\(\begin{aligned} \mathrm{I} &=\int_{2}^{3} \frac{\mathrm{dx}}{\mathrm{x}^{2}+\mathrm{x}}=\int_{2}^{3} \frac{\mathrm{dx}}{\mathrm{x}(\mathrm{x}+1)} \\ \therefore \mathrm{I} &=\int_{2}^{3}\left[\frac{1}{\mathrm{x}}-\frac{1}{\mathrm{x}+1}\right] \mathrm{dx} \\ &=[\log \mathrm{x}-\log (\mathrm{x}+1)]_{2}^{3}=\left[\log \left(\frac{\mathrm{x}}{\mathrm{x}+1}\right)\right]_{2}^{3} \\ &=\log \left(\frac{3}{4}\right)-\log \left(\frac{2}{3}\right)=\log \left(\frac{3}{4} \times \frac{3}{2}\right)=\log \left(\frac{9}{8}\right) \end{aligned}\)
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