NCERT Class 12 · Ch 7 Exercise 7.2 · Q37
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.2 Question 37
Evaluate \(\int \frac{x^{7}}{\left(x^{8} + 1\right)^{\frac{3}{2}}}\, dx\).
💡 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: \(- \frac{1}{4 \sqrt{x^{8} + 1}}\) + C.
- Work on each open interval where the original integrand is real and defined.
- u=tan⁻¹(x⁴), du=4x³dx/(1+x⁸).
- An antiderivative is F(x)=\(- \frac{1}{4 \sqrt{x^{8} + 1}}\).
- Differentiate this expression to check that F′(x)=\(\frac{x^{7}}{\left(x^{8} + 1\right)^{\frac{3}{2}}}\). Add an arbitrary constant C on each domain component.
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