NCERT Class 12 · Ch 7 Miscellaneous Exercise · Q16
NCERT Class 12 Maths · Chapter 7 Integrals · Miscellaneous Exercise Question 16
Evaluate \(\int \frac{x^{3}}{x^{4} + 1}\, dx\). Original logarithm domain: x>0.
💡 Concept Hints
- Identify the domain and the inner and outer functions, or the relevant one-sided limits.
- Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.
✅ Step-by-Step Solution
Final Answer: \(\frac{\log{\left(x^{4} + 1 \right)}}{4}\) + C.
- Work on each open interval where the original integrand is real and defined.
- On x>0 this is x³/(x⁴+1). Put u=x⁴+1.
- An antiderivative is F(x)=\(\frac{\log{\left(x^{4} + 1 \right)}}{4}\).
- Differentiate this expression to check that F′(x)=\(\frac{x^{3}}{x^{4} + 1}\). Add an arbitrary constant C on each domain component.
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