TS EAMCET · Maths · Indefinite Integration
\(\int \frac{d x}{(x+1) \sqrt{x^2+4}}=\)
- A \(\frac{1}{2} \sqrt{\frac{x+1}{x+2}}+c\)
- B \(\log \left|\frac{x+2}{x+1}\right|+c\)
- C \(-\frac{1}{\sqrt{5}} \sinh ^{-1}\left(\frac{4-x}{2(x+1)}\right)+c\)
- D \(-\frac{1}{\sqrt{5}} \cosh ^{-1}\left(\frac{4+x}{2(x-1)}\right)+c\)
Answer & Solution
Correct Answer
(C) \(-\frac{1}{\sqrt{5}} \sinh ^{-1}\left(\frac{4-x}{2(x+1)}\right)+c\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & \text { Let } K=\int \frac{d x}{(x+1) \sqrt{x^2+4}} \\ & \text { Let } x+1=\frac{1}{t} \Rightarrow x=\frac{1}{t}-1=\frac{1-t}{t} \Rightarrow d x=-\frac{1}{t^2} d t \\ & I=-\int \frac{\frac{1}{t^2} d t}{\frac{1}{t} \sqrt{\left(\frac{1-t}{t}\right)^2+4}}=-\int…
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