Class 12 Integrals · Long Answer3 Marksindefinite_integration
Class 12 Maths · Integrals · Long Answer #152
Evaluate \(\int \frac{1}{x \left(x^{5} + 3\right)}\,dx\). Work on each connected open interval where the real integrand is defined.
✅ Step-by-Step Solution
Final Answer: \(\frac{\log{\left(x^{4} \left|{x}\right| \left|{\frac{1}{x^{5} + 3}}\right| \right)}}{15}\) + C
- Let u=x⁵, so dx/x=du/(5u) on an interval avoiding x=0.
- The integral is (1/5)∫du/[u(u+3)]=(1/15)∫[1/u−1/(u+3)]du.
- Integrate and substitute u=x⁵ to obtain (1/15)ln|x⁵/(x⁵+3)|.
- Therefore the required integral is \(\frac{\log{\left(x^{4} \left|{x}\right| \left|{\frac{1}{x^{5} + 3}}\right| \right)}}{15}\) + C.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Let u=x⁵, so dx/x=du/(5u) on an interval avoiding x=0.
- [1 mark] The integral is (1/5)∫du/[u(u+3)]=(1/15)∫[1/u−1/(u+3)]du.
- [1 mark] Integrate and substitute u=x⁵ to obtain (1/15)ln|x⁵/(x⁵+3)|.
- [0.5 marks] Therefore the required integral is \(\frac{\log{\left(x^{4} \left|{x}\right| \left|{\frac{1}{x^{5} + 3}}\right| \right)}}{15}\) + C.
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