COMEDK · Maths · 28. Indefinite Integration
\(\int \frac{x^{3}-1}{x^{3}+x} d x=\)
- A \(x-\log x+\log \left(x^{2}+1\right)-\tan ^{-1} x+c\)
- B \(x-\log x+\frac{1}{2} \log \left(x^{2}+1\right)-\tan ^{-1} x+c\)
- C \(x+\log x+\log \left(x^{2}+1\right)-\tan ^{-1} x+c\)
- D \(x+\log x+\frac{1}{2} \log \left(x^{2}+1\right)-\tan ^{-1} x+c\)
Answer & Solution
Correct Answer
(B) \(x-\log x+\frac{1}{2} \log \left(x^{2}+1\right)-\tan ^{-1} x+c\)
Step-by-step Solution
Detailed explanation
Let \[ \begin{aligned} I &=\int \frac{x^{3}-1}{x^{3}+x} d x=\int\left(1-\frac{x+1}{x^{3}+x}\right) d x \\ &=\int 1 d x-\int \frac{x+1}{x\left(x^{2}+1\right)} d x=x-\int \frac{x+1}{x\left(x^{2}+1\right)} d x \ldots(\mathrm{i}) \end{aligned} \] Now,…
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