WBJEE · Maths · Indefinite Integration
\(\int \frac{\mathrm{f}(\mathrm{x}) \varphi^{\prime}(\mathrm{x})+\varphi(\mathrm{x}) \mathrm{f}^{\prime}(\mathrm{x})}{(\mathrm{f}(\mathrm{x}) \varphi(\mathrm{x})+1) \sqrt{\mathrm{f}(\mathrm{x}) \varphi(\mathrm{x})-1}} \mathrm{dx}=\)
- A \(\sin ^{-1} \sqrt{\frac{\mathrm{f}(\mathrm{x})}{\varphi(\mathrm{x})}}+\mathrm{c}\)
- B \(\cos ^{-1} \sqrt{(f(x))^{2}-(\varphi(x))^{2}}+c\)
- C \(\sqrt{2} \tan ^{-1} \sqrt{\frac{f(x) \varphi(x)-1}{2}}+c\)
- D \(\sqrt{2} \tan ^{-1} \sqrt{\frac{f(x) \varphi(x)+1}{2}}+c\)
Answer & Solution
Correct Answer
(C) \(\sqrt{2} \tan ^{-1} \sqrt{\frac{f(x) \varphi(x)-1}{2}}+c\)
Step-by-step Solution
Detailed explanation
Hint: Let \(f(x) \phi(x)=t\) \(\int \frac{d t}{(t+1) \sqrt{t-1}}\) Let \(t-1=p^{2}, d t=2 p d p\) \(\Rightarrow \int \frac{2 d p}{p^{2}+2}=\sqrt{2} \tan ^{-1}\left|\sqrt{\frac{f(x) \phi(x)-1}{2}}\right|+c\)
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