NCERT Class 12 · Ch 7 Miscellaneous Exercise · Q4
NCERT Class 12 Maths · Chapter 7 Integrals · Miscellaneous Exercise Question 4
Evaluate \(\int \frac{1}{x^{2} \left(x^{4} + 1\right)^{\frac{3}{4}}}\, dx\).
💡 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: \(- \frac{\sqrt[4]{x^{4} + 1}}{x}\) + C.
- Work on each open interval where the original integrand is real and defined.
- Use u=1/x: the integrand transforms into −u³/(u⁴+1)^(3/4)du on x>0; alternatively differentiate the proposed real fourth-root quotient on either sign component.
- An antiderivative is F(x)=\(- \frac{\sqrt[4]{x^{4} + 1}}{x}\).
- Differentiate this expression to check that F′(x)=\(\frac{1}{x^{2} \left(x^{4} + 1\right)^{\frac{3}{4}}}\). Add an arbitrary constant C on each domain component.
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