TS EAMCET · Maths · Indefinite Integration
\(\int \frac{1}{(x+2) \sqrt{x^2+x+2}} d x=\)
- A \(-\frac{1}{2} \operatorname{Sinh}^{-1}\left(\frac{2-3 x}{\sqrt{7}(x+2)}\right)+\mathrm{c}\)
- B \(-\frac{1}{2} \operatorname{Sin}^{-1}\left(\frac{2+3 x}{\sqrt{7}(x+2)}\right)+\mathrm{c}\)
- C \(\frac{1}{2} \operatorname{Cosh}^{-1}\left(\frac{2+3 x}{\sqrt{7}(x+2)}\right)+\mathrm{c}\)
- D \(-\frac{1}{2} \operatorname{Cos}^{-1}\left(\frac{2-3 x}{\sqrt{7}(x+2)}\right)+\mathrm{c}\)
Answer & Solution
Correct Answer
(A) \(-\frac{1}{2} \operatorname{Sinh}^{-1}\left(\frac{2-3 x}{\sqrt{7}(x+2)}\right)+\mathrm{c}\)
Step-by-step Solution
Detailed explanation
Let \(x+2 = \frac{1}{t} \implies x = \frac{1}{t}-2, dx = -\frac{1}{t^2} dt\). \(\int \frac{1}{(x+2) \sqrt{x^2+x+2}} d x = \int \frac{t}{\sqrt{(\frac{1}{t}-2)^2 + (\frac{1}{t}-2) + 2}} (-\frac{1}{t^2}) dt\) \(= -\int \frac{1}{t \sqrt{\frac{1}{t^2} - \frac{3}{t} + 4}} dt\)…
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