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