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Class 12 Integrals · Very Long Answer5 Marksindefinite_integration

Class 12 Maths · Integrals · Very Long Answer #174

Evaluate \(\int \frac{1}{x \left(x^{4} - 1\right)}\,dx\). Work on each connected open interval where the real integrand is defined.

✅ Step-by-Step Solution

Final Answer: \(\frac{\log{\left(\left|{\frac{x^{4} - 1}{x^{4}}}\right| \right)}}{4}\) + C
  1. Let u=x⁴, so dx/x=du/(4u) on each interval avoiding x=0.
  2. The integral becomes (1/4)∫du/[u(u−1)].
  3. Decompose 1/[u(u−1)]=−1/u+1/(u−1).
  4. Integrate to (1/4)[ln|u−1|−ln|u|] and substitute u=x⁴.
  5. Therefore the required integral is \(\frac{\log{\left(\left|{\frac{x^{4} - 1}{x^{4}}}\right| \right)}}{4}\) + C.

📋 CBSE Stepwise Marking Scheme

  • [1 mark] Let u=x⁴, so dx/x=du/(4u) on each interval avoiding x=0.
  • [1 mark] The integral becomes (1/4)∫du/[u(u−1)].
  • [1 mark] Decompose 1/[u(u−1)]=−1/u+1/(u−1).
  • [1 mark] Integrate to (1/4)[ln|u−1|−ln|u|] and substitute u=x⁴.
  • [1 mark] Therefore the required integral is \(\frac{\log{\left(\left|{\frac{x^{4} - 1}{x^{4}}}\right| \right)}}{4}\) + C.

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