TS EAMCET · Maths · Indefinite Integration
\(\int \frac{x^3}{x^4+3 x^2+2} d x=\)
- A \(\log \left(\frac{x^2+2}{\sqrt{x^2+1}}\right)+\mathrm{c}\)
- B \(\log \left(x^2+2\right)-2 \log \left(x^2+1\right)+\mathrm{c}\)
- C \(\log \left(\frac{\left(x^2+2\right) x}{\sqrt{x^2+1}}\right)+\mathrm{c}\)
- D \(\log \left(\frac{x^2+1}{\sqrt{x^2+2}}\right)+\mathrm{c}\)
Answer & Solution
Correct Answer
(A) \(\log \left(\frac{x^2+2}{\sqrt{x^2+1}}\right)+\mathrm{c}\)
Step-by-step Solution
Detailed explanation
Let \(u = x^2\). \(du = 2x \, dx \implies x \, dx = \frac{1}{2} du\) \(\int \frac{x^3}{x^4+3 x^2+2} d x = \frac{1}{2} \int \frac{u}{u^2+3u+2} du\) \(= \frac{1}{2} \int \frac{u}{(u+1)(u+2)} du\) Using partial fractions: \(\frac{u}{(u+1)(u+2)} = \frac{2}{u+2} - \frac{1}{u+1}\)…
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