2025 RI P1 Q4
2025 RI P1 Q4
10 marks
- Find \(\displaystyle \displaystyle\int \frac{9x}{(2x - 1)(x + 1)^2} \, \mathrm{d}x\).[4]
- Perform the following differentiations:
- Differentiate \(\frac{1}{x^2 + 1}\) with respect to \(x\).[1]
- Differentiate \(\ln \sqrt{x^2 + 1}\) with respect to \(x\).[1]
- Hence find \(\displaystyle \displaystyle\int \frac{x \ln \sqrt{x^2 + 1}}{(x^2 + 1)^2} \, \mathrm{d}x\).[4]
Solution:
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Answer:(a) \(\ln|2x-1|-\ln|x+1|-\frac{3}{x+1}+C\). (b)(i) \(-\frac{2x}{(x^2+1)^2}\). (ii) \(\frac{x}{x^2+1}\). (iii) \(-\frac{\ln(x^2+1)+1}{4(x^2+1)}+C\).
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