CUET · MATHS · PYQ PAPER 2025
If \(2 f(x)+f\left(\frac{1}{x}\right)=x^2+1\), then \(\int f(x) d x\) is : (Here \(C\) is an arbitrary constant)
- A \(\frac{1}{3}\left(\frac{2}{3} x^3+\frac{1}{x}+x\right)+C\)
- B \(\frac{2}{3} x^3-\frac{1}{x}+x+C\)
- C \(\frac{x^3}{3}+x+C\)
- D \(\frac{1}{3}\left(\frac{2}{3} x^3-x\right)+C\)
Answer & Solution
Correct Answer
(A) \(\frac{1}{3}\left(\frac{2}{3} x^3+\frac{1}{x}+x\right)+C\)
Step-by-step Solution
Detailed explanation
\(2 f(x)+f\left(\frac{1}{x}\right)=x^2+1 \quad (1)\) \(2 f\left(\frac{1}{x}\right)+f(x)=\frac{1}{x^2}+1 \quad (2)\) \((1) \times 2 - (2) \implies 3 f(x) = 2(x^2+1) - \left(\frac{1}{x^2}+1\right) = 2x^2+2-\frac{1}{x^2}-1 = 2x^2+1-\frac{1}{x^2}\)…
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