✓BeMath✓ (1) ∫_0 ^∞ ((√x)/(1+x^3 )) dx ? (2) lim_(x→0) ((sin (π cos^2 x))/x^2 ) (3) If g(x)= 1+(√x) and (g○f)(x)=3+2(√x) +x find f(x)


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Question Number 107706 by bemath last updated on 12/Aug/20

$✓ B e M a t h ✓$ $(1) \underset{0}{\int^{\infty}} \frac{\sqrt{x}}{1 + x^{3}} d x ?$ $(2) \lim_{x \to 0} \frac{\sin (π \cos^{2} x)}{x^{2}}$ $(3) I f g (x) = 1 + \sqrt{x} a n d (g \circ f) (x) = 3 + 2 \sqrt{x} + x$ $f i n d f (x)$
	Answered by bemath last updated on 12/Aug/20
	$(3) (g○f)(x)= 1+(√(f(x))) =x+2(√x) +3 (√(f(x))) = x+2(√x) +2 (√(f(x))) = ((√x))^2 + 2(√x) +2 (√(f(x))) = ((√x) +1)^2 +1 f(x) = { ((√x)+1)^2 +1}^2$
	$(3) (g \circ f) (x) = 1 + \sqrt{f (x)} = x + 2 \sqrt{x} + 3$ $\sqrt{f (x)} = x + 2 \sqrt{x} + 2$ $\sqrt{f (x)} = {(\sqrt{x})}^{2} + 2 \sqrt{x} + 2$ $\sqrt{f (x)} = {(\sqrt{x} + 1)}^{2} + 1$ $f (x) = {{(\sqrt{x} + 1)}^{2} + 1}^{2}$
	Answered by Dwaipayan Shikari last updated on 12/Aug/20

	$\lim_{x \to 0} \frac{s i n (π {c o s}^{2} x)}{x^{2}} = \frac{s i n (π - π {s i n}^{2} x)}{x^{2}} = \frac{s i n π c o s (π {s i n}^{2} x) - c o s π s i n (π {s i n}^{2} x)}{x^{2}}$ $= \frac{s i n (π {s i n}^{2} x)}{x^{2}} = \frac{s i n (π x^{2})}{x^{2}} = \frac{π x^{2}}{x^{2}} = π (s i n (π {s i n}^{2} x) \to s i n (π x^{2}))$ $(s i n (π x^{2}) \to π x^{2})$
	Answered by Dwaipayan Shikari last updated on 12/Aug/20

	$\lim_{x \to 0} \frac{s i n (π {c o s}^{2} x)}{x^{2}} = π \frac{c o s (π {c o s}^{2} x)}{2 x} - 2 c o s x s i n x = π \frac{c o s (π {c o s}^{2} x)}{2 x} - 2 x$ $A n o t h e r w a y$ $= π (- 1) (- 1) = π$
	Answered by john santu last updated on 12/Aug/20

	$\frac{♣ JS ♣}{\dots}$ $(1) \int {\overset{\infty}{}}_{0} \frac{\sqrt{x}}{1 + x^{3}} d x [l e t h = \sqrt{x}]$ $\underset{0}{\int^{\infty}} \frac{h}{1 + h^{6}} . (2 h) d h = \underset{0}{\int^{\infty}} \frac{2 h^{2}}{1 + h^{6}} d h$ $n o w s e t q = h^{3};$ $\underset{0}{\int^{\infty}} \frac{2}{1 + q^{3}} . \frac{1}{3} d q = {[\frac{2}{3} arc \tan q]}_{0}^{\infty}$ $= \frac{2}{3} \times \frac{π}{2} = = \frac{π}{3}$
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