AP EAMCET · Maths · Indefinite Integration
\(\int \frac{x^3-1}{x^3+x} d x=\)
- A \(x+\log |x|+\frac{1}{2} \log \left(x^2+1\right)+\sin ^{-1}(x)+c\)
- B \(x-\log |x|+\frac{1}{2} \log \left(x^2+1\right)-\sin ^{-1}(x)+c\)
- C \(x+\log |x|-\frac{1}{2} \log \left(x^2+1\right)+\tan ^{-1}(x)+c\)
- D \(x-\log |x|+\frac{1}{2} \log \left(x^2+1\right)-\tan ^{-1}(x)+c\)
Answer & Solution
Correct Answer
(D) \(x-\log |x|+\frac{1}{2} \log \left(x^2+1\right)-\tan ^{-1}(x)+c\)
Step-by-step Solution
Detailed explanation
\(\begin{aligned} & I=\int\left(\frac{x^3-1}{x^3+x}\right) d x=\int\left(1-\frac{x+1}{x^3+x}\right) d x \\ & \Rightarrow \int 1 \cdot d x-\int \frac{(x+1)}{x^3+x} d x=x-\int \frac{(x+1)}{x\left(x^2+1\right)} d x \end{aligned}\) Now,…
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