BeMath-1-0-x-1-x-3-dx-2-lim-x-0-sin-pi-cos-2-x-x-2-3-If-g-x-1-x-and-g-f-x-3-2-x-x-find-f-x-

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 \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}