NCERT Class 12 · Ch 7 Miscellaneous Exercise · Q5
NCERT Class 12 Maths · Chapter 7 Integrals · Miscellaneous Exercise Question 5
Evaluate \(\int \frac{1}{\sqrt[3]{x} + \sqrt{x}}\, dx\).
💡 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: \(6 \sqrt[6]{x} - 3 \sqrt[3]{x} + 2 \sqrt{x} - 6 \log{\left(\left|{\sqrt[6]{x} + 1}\right| \right)}\) + C.
- Work on each open interval where the original integrand is real and defined.
- Put t=x^(1/6)>0: the integral is ∫6t³/(t+1)dt=∫6(t²−t+1−1/(t+1))dt.
- An antiderivative is F(x)=\(6 \sqrt[6]{x} - 3 \sqrt[3]{x} + 2 \sqrt{x} - 6 \log{\left(\left|{\sqrt[6]{x} + 1}\right| \right)}\).
- Differentiate this expression to check that F′(x)=\(\frac{1}{\sqrt[3]{x} + \sqrt{x}}\). Add an arbitrary constant C on each domain component.
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