Dividing first, then recognising a derivative
Dividing first, then recognising a derivative
IP 4
7 marks
- Express \(\dfrac{x^2+3x+5}{x+1}\) in the form \(Ax+B+\dfrac{C}{x+1}\).[2]
- Hence find \(\displaystyle\int\dfrac{x^2+3x+5}{x+1}\,\mathrm dx\) for \(x>-1\).[2]
- Find \(\displaystyle\int\dfrac{2x+3}{x^2+3x+5}\,\mathrm dx\), and explain why your answer is valid for all real \(x\).[3]
Solution:
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Answer:(a) \(x+2+\dfrac{3}{x+1}\) (b) \(\dfrac{x^2}{2}+2x+3\ln(x+1)+C\) (c) \(\ln\left(x^2+3x+5\right)+C\); \(x^2+3x+5>0\) for all \(x\)
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