Question

$$\int \frac{f(x) \varphi^{\prime}(x)+\varphi(x) f^{\prime}(x)}{(f(x) \varphi(x)+1) \sqrt{f(x) \varphi(x)-1}} d x=$$
(A) $\sin ^{-1} \sqrt{\frac{f(x)}{\varphi(x)}}+c$
(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 1

Ans: (C)
Hint : Let $f(x) \phi(x)=t$
$\int \frac{d t}{(t+1) \sqrt{t-1}}$ Let $t-1=p^{2}, d t=2 p d p$ $\Rightarrow \int \frac{2 d p}{p^{2}+2}=\sqrt{2} \tan ^{-1}\left|\sqrt{\frac{f(x) \phi(x)-1}{2}}\right|+c$

Answer 2

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Answer 3

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