The value of $ \int \frac{f(x) \phi^{\prime}(x)+\phi(x) f^{\prime}(x)}{(f(x) \cdot \phi(x)+1) \sqrt{\phi(x) \cdot f(x)-1}} d x $ is (Where $ C $ is the constant of integration)

Question

The value of $ \int \frac{f(x) \phi^{\prime}(x)+\phi(x) f^{\prime}(x)}{(f(x) \cdot \phi(x)+1) \sqrt{\phi(x) \cdot f(x)-1}} d x $ is (Where $ C $ is the constant of integration), $ \cos ^{-1} \sqrt{f(x)^{2}-\phi(x)^{2}} $, $ \tan ^{-1}[f(x) \phi(x)] $, $ \sin ^{-1} \sqrt{\frac{f(x)}{\phi(x)}} $, None of these

MARKS App · Accepted Answer

Let I = ∫ f ( x ) ϕ ' ( x ) + ϕ ( x ) f ' ( x ) f ( x ) · ϕ ( x ) + 1 ϕ ( x ) · f ( x ) - 1 d x Let ϕ ( x ) · f ( x ) - 1 = t 2 ⇒ ( ϕ ( x ) · f ' ( x ) + f ( x ) ϕ ' ( x ) ) d x = 2 t d t ∴ I = ∫ 2 t d t ( t 2 + 2 ) t 2 = ∫ 2 d t t 2 + ( 2 ) 2 = 2 · 1 2 tan - 1 t 2 + C = 2 tan - 1 ϕ ( x ) f ( x ) - 1 2 + C

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