AP EAMCET · Maths · Indefinite Integration
\(\int \frac{(x+1)^2}{x\left(x^2+1\right)} d x=\)
- A \(\log \left[x\left(x^2+1\right)\right]+c\)
- B \(\log |x|+c\)
- C \(\log |x|+2 \tan ^{-1}(x)+c\)
- D \(2 \log |x|+\tan ^{-1}(x)+c\)
Answer & Solution
Correct Answer
(C) \(\log |x|+2 \tan ^{-1}(x)+c\)
Step-by-step Solution
Detailed explanation
\(\begin{aligned} I & =\int \frac{(x+1)^2}{x\left(x^2+1\right)} d x=\int \frac{\left(x^2+1\right)+2 x}{x\left(x^2+1\right)} d x \\ & =\int \frac{1}{x} d x+2 \int \frac{d x}{x^2+1}=\log _e|x|+2 \tan ^{-1} x+C \end{aligned}\) Hence, option (c) is correct.
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