Class 12 Integrals · Long Answer3 Marksindefinite_integration
Class 12 Maths · Integrals · Long Answer #154
Evaluate \(\int \frac{1}{x \left(x^{3} + 8\right)}\,dx\). Work on each connected open interval where the real integrand is defined.
✅ Step-by-Step Solution
Final Answer: \(\frac{\log{\left(x^{2} \left|{x}\right| \left|{\frac{1}{x^{3} + 8}}\right| \right)}}{24}\) + C
- Set u=x³, so dx/x=du/(3u).
- Then I=(1/3)∫du/[u(u+8)]=(1/24)∫[1/u−1/(u+8)]du.
- Integrate the two reciprocals and substitute u=x³.
- Therefore the required integral is \(\frac{\log{\left(x^{2} \left|{x}\right| \left|{\frac{1}{x^{3} + 8}}\right| \right)}}{24}\) + C.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Set u=x³, so dx/x=du/(3u).
- [1 mark] Then I=(1/3)∫du/[u(u+8)]=(1/24)∫[1/u−1/(u+8)]du.
- [1 mark] Integrate the two reciprocals and substitute u=x³.
- [0.5 marks] Therefore the required integral is \(\frac{\log{\left(x^{2} \left|{x}\right| \left|{\frac{1}{x^{3} + 8}}\right| \right)}}{24}\) + C.
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