MHT CET · Maths · Indefinite Integration
The value of \(\int \frac{2 x^3-1}{x^4+x} \mathrm{~d} x\) is equal to (where \(C\) is a constant of integration.)
- A \(\frac{1}{2} \log \frac{\left(x^3+1\right)^2}{x^3}+C\)
- B \(\log \frac{\left(x^3+1\right)}{x}+C\)
- C \(\log \left(\frac{x^3+1}{x^2}\right)+C\)
- D \(\frac{1}{2} \log \frac{\left(x^3+1\right)}{x^2}+C\)
Answer & Solution
Correct Answer
(B) \(\log \frac{\left(x^3+1\right)}{x}+C\)
Step-by-step Solution
Detailed explanation
\(\int \frac{2 x^3-1}{x^4+x} \mathrm{~d} x=\int \frac{2 x^3-1}{x\left(x^3+1\right)} \mathrm{d} x=\) \(\int\left(\frac{3 x^2}{x^3+1}-\frac{1}{x}\right) \mathrm{d} x\)
\(=\log \left(x^3+1\right)-\log x+c\)
\(=\log \left(\frac{x^3+1}{x}\right)+c\)
\(=\log \left(x^3+1\right)-\log x+c\)
\(=\log \left(\frac{x^3+1}{x}\right)+c\)
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