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N2012 P1 Q2

N2012 P1 Q2

Junior College 2
6 marks
  1. Find \(\displaystyle\int\frac{x^3}{1+x^4}\,\mathrm dx\).[2]
  2. Use the substitution \(u=x^2\) to find \(\displaystyle\int\frac{x}{1+x^4}\,\mathrm dx\).[3]
  3. Evaluate \(\displaystyle\int_0^1\left(\frac{x}{1+x^4}\right)^2\,\mathrm dx\), giving the answer correct to 3 decimal places.[1]

Solution:

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Answer:(i) \(\frac14\ln(1+x^4)+C\). (ii) \(\frac12\tan^{-1}(x^2)+C\). (iii) \(0.186\).

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