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

viagra cialis <a href="https://ordergnonline.com/">order cialis 20mg</a> buy ed pills canada

Answer 3

cialis 5mg <a href="https://ordergnonline.com/">buy generic cialis</a> buying ed pills online