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Question Number 55273 Answers: 0 Comments: 1

let f(x) =∫_x^2 ^(1+x) (dt/(1+t+t^2 )) 1) calculate f(x) interms of x 2) calculate lim_(x→0) f(x) and lim_(x→+∞) f(x)

$l e t f (x) = \int_{x^{2}}^{1 + x} \frac{d t}{1 + t + t^{2}}$ $1) c a l c u l a t e f (x) i n t e r m s o f x$ $2) c a l c u l a t e {l i m}_{x \to 0} f (x) a n d {l i m}_{x \to + \infty} f (x)$

Question Number 55271 Answers: 0 Comments: 0

1) let f(x) =∫_0 ^(2π) ((cost)/(3 +sin(xt)))dt find a explicit form of f(x) 2) calculate g(x) =∫_0 ^(2π) ((tcos(xt)cost)/((3 +sin(xt))^2 ))dt 3) calculate ∫_0 ^(2π) ((cost)/(3+sint)) and ∫_0 ^(2π) ((t cos^2 t)/((3+sint)^2 ))dt

$1) l e t f (x) = \int_{0}^{2 π} \frac{c o s t}{3 + s i n (x t)} d t$ $f i n d a e x p l i c i t f o r m o f f (x)$ $2) c a l c u l a t e g (x) = \int_{0}^{2 π} \frac{t c o s (x t) c o s t}{{(3 + s i n (x t))}^{2}} d t$ $3) c a l c u l a t e \int_{0}^{2 π} \frac{c o s t}{3 + s i n t} a n d \int_{0}^{2 π} \frac{t {c o s}^{2} t}{{(3 + s i n t)}^{2}} d t$

Question Number 55270 Answers: 0 Comments: 1

let f(x)=x^n arctan(x^2 ) with n integr natural 1) calculate f^((n)) (x) and f^((n)) (0) 2)developp f at integr serie .

$l e t f (x) = x^{n} a r c t a n (x^{2}) w i t h n i n t e g r n a t u r a l$ $1) c a l c u l a t e f^{(n)} (x) a n d f^{(n)} (0)$ $2) d e v e l o p p f a t i n t e g r s e r i e .$

Question Number 55269 Answers: 0 Comments: 4

let f(x) =(2x+1)ln(1−x^2 ) 1) calculate f^((n)) (x) and f^((n)) (0) 2)developp f at integr serie. 3) calculate ∫_0 ^1 f(x)dx .

$l e t f (x) = (2 x + 1) l n (1 - x^{2})$ $1) c a l c u l a t e f^{(n)} (x) a n d f^{(n)} (0)$ $2) d e v e l o p p f a t i n t e g r s e r i e .$ $3) c a l c u l a t e \int_{0}^{1} f (x) d x .$

Question Number 55268 Answers: 0 Comments: 1

calculate Σ_(n=0) ^∞ (((−1)^n )/(4n+3))

$c a l c u l a t e \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{4 n + 3}$

Question Number 55308 Answers: 1 Comments: 0

Question Number 55288 Answers: 2 Comments: 0

Question Number 55258 Answers: 0 Comments: 0

3x+5y=?_

$3 x + 5 y = ?_{}$

Question Number 55267 Answers: 0 Comments: 0

1) calculate f(x)=∫_0 ^(π/4) ln(1+xtanθ)dθ 2) find the values of integrals ∫_0 ^(π/4) ln(1+tanθ) and ∫_0 ^(π/4) ln(1+2tanθ)dθ . 1) we have f^′ (x)=∫_0 ^(π/4) ((tanθ)/(1+xtanθ)) dθ =∫_0 ^(π/4) (((sinθ)/(cosθ))/(1+x((sinθ)/(cosθ))))dθ =∫_0 ^(π/4) ((sinθ)/(cosθ +xsinθ)) dθ =_(tan((θ/2))=t) ∫_0 ^((√2)−1) (((2t)/(1+t^2 ))/(((1−t^2 )/(1+t^2 )) +((2xt)/(1+t^2 )))) ((2dt)/(1+t^2 )) =∫_0 ^((√2)−1) ((4t)/((1+t^2 )(1−t^2 +2xt)))dt =−∫_0 ^((√2)−1) ((4t)/((t^2 +1)(t^2 −2xt −1)))dt let decompose F(t) = ((4t)/((t^2 +1)(t^2 −2xt −1))) roots of t^2 −2xt −1 Δ^′ =x^2 +1 ⇒t_1 =x+(√(x^2 +1)) and t_2 =x−(√(x^2 +1)) F(t)=(a/(t−t_1 )) +(b/(t−t_2 )) +((ct +d)/(t^2 +1)) a =lim_(t→t_1 ) (t−t_1 )F(t)=((4t_1 )/((t_1 ^2 +1)(t_1 −t_2 ))) =α b =lim_(t→t_2 ) (t−t_2 )F(t) =((4t_2 )/((t_2 ^2 +1)(t_2 −t_1 ))) =β ⇒F(t)=(α/(t−t_1 )) +(β/(t−t_2 )) +((ct +d)/(t^2 +1)) F(0) =0=−(α/t_1 ) −(β/t_2 ) +d ⇒d =(α/t_1 ) +(β/t_2 ) F(1)=(2/(−2x)) =−(1/x)=(α/(1−t_1 )) +(β/(1−t_2 )) +((c+d)/2) ⇒(1/x) =(α/(t_1 −1)) +(β/(t_2 −1)) −(c/2) −(d/2) ⇒(c/2) =(α/(t_1 −1)) +(β/(t_2 −1)) −(d/2) −(1/x) ⇒c =((2α)/(t_1 −1)) +((2β)/(t_2 −1)) −d−(2/x) ∫ F(t)dt =αln∣t−t_1 ∣ +βln∣t−t_2 ∣ +(c/2)ln(t^2 +1) +d arctan(t) ⇒ ∫_0 ^((√2)−1) F(t)dt =[αln∣t−t_1 ∣+βln∣t−t_2 ∣ +(c/2)ln(t^2 +1)]_0 ^((√2)−1) =αln∣(√2)−1−t_1 ∣ +βln∣(√2)−1−t_2 ∣ +(c/2)ln(4−2(√2)) =αln∣(√2)−1−x−(√(1+x^2 )))+βln∣(√2)−1−x+(√(1+x^2 ))) +((ln(4−2(√2)))/2)c =f^′ (x) ⇒ f(x)=∫ αln∣(√2)−1−x−(√(1+x^2 ))∣)dx+β∫ ln∣(√2)−1+(√(1+x^2 ))∣dx +((cx)/2)ln(4−2(√2)) +C ....be continued...

$1) c a l c u l a t e f (x) = \int_{0}^{\frac{π}{4}} l n (1 + x t a n θ) d θ$ $2) f i n d t h e v a l u e s o f i n t e g r a l s \int_{0}^{\frac{π}{4}} l n (1 + t a n θ) a n d \int_{0}^{\frac{π}{4}} l n (1 + 2 t a n θ) d θ .$ $1) w e h a v e f^{^{'}} (x) = \int_{0}^{\frac{π}{4}} \frac{t a n θ}{1 + x t a n θ} d θ = \int_{0}^{\frac{π}{4}} \frac{\frac{s i n θ}{c o s θ}}{1 + x \frac{s i n θ}{c o s θ}} d θ$ $= \int_{0}^{\frac{π}{4}} \frac{s i n θ}{c o s θ + x s i n θ} d θ =_{t a n (\frac{θ}{2}) = t} \int_{0}^{\sqrt{2} - 1} \frac{\frac{2 t}{1 + t^{2}}}{\frac{1 - t^{2}}{1 + t^{2}} + \frac{2 x t}{1 + t^{2}}} \frac{2 d t}{1 + t^{2}}$ $= \int_{0}^{\sqrt{2} - 1} \frac{4 t}{(1 + t^{2}) (1 - t^{2} + 2 x t)} d t = - \int_{0}^{\sqrt{2} - 1} \frac{4 t}{(t^{2} + 1) (t^{2} - 2 x t - 1)} d t l e t d e c o m p o s e$ $F (t) = \frac{4 t}{(t^{2} + 1) (t^{2} - 2 x t - 1)} r o o t s o f t^{2} - 2 x t - 1$ $Δ^{^{'}} = x^{2} + 1 \Rightarrow t_{1} = x + \sqrt{x^{2} + 1} a n d t_{2} = x - \sqrt{x^{2} + 1}$ $F (t) = \frac{a}{t - t_{1}} + \frac{b}{t - t_{2}} + \frac{c t + d}{t^{2} + 1}$ $a = {l i m}_{t \to t_{1}} (t - t_{1}) F (t) = \frac{4 t_{1}}{(t_{1}^{2} + 1) (t_{1} - t_{2})} = α$ $b = {l i m}_{t \to t_{2}} (t - t_{2}) F (t) = \frac{4 t_{2}}{(t_{2}^{2} + 1) (t_{2} - t_{1})} = β \Rightarrow F (t) = \frac{α}{t - t_{1}} + \frac{β}{t - t_{2}} + \frac{c t + d}{t^{2} + 1}$ $F (0) = 0 = - \frac{α}{t_{1}} - \frac{β}{t_{2}} + d \Rightarrow d = \frac{α}{t_{1}} + \frac{β}{t_{2}}$ $F (1) = \frac{2}{- 2 x} = - \frac{1}{x} = \frac{α}{1 - t_{1}} + \frac{β}{1 - t_{2}} + \frac{c + d}{2} \Rightarrow \frac{1}{x} = \frac{α}{t_{1} - 1} + \frac{β}{t_{2} - 1} - \frac{c}{2} - \frac{d}{2}$ $\Rightarrow \frac{c}{2} = \frac{α}{t_{1} - 1} + \frac{β}{t_{2} - 1} - \frac{d}{2} - \frac{1}{x} \Rightarrow c = \frac{2 α}{t_{1} - 1} + \frac{2 β}{t_{2} - 1} - d - \frac{2}{x}$ $\int F (t) d t = α l n ∣ t - t_{1} ∣ + β l n ∣ t - t_{2} ∣ + \frac{c}{2} l n (t^{2} + 1) + d a r c t a n (t) \Rightarrow$ $\int_{0}^{\sqrt{2} - 1} F (t) d t = {[α l n ∣ t - t_{1} ∣ + β l n ∣ t - t_{2} ∣ + \frac{c}{2} l n (t^{2} + 1)]}_{0}^{\sqrt{2} - 1}$ $= α l n ∣ \sqrt{2} - 1 - t_{1} ∣ + β l n ∣ \sqrt{2} - 1 - t_{2} ∣ + \frac{c}{2} l n (4 - 2 \sqrt{2})$ $= α l n ∣ \sqrt{2} - 1 - x - \sqrt{1 + x^{2}}) + β l n ∣ \sqrt{2} - 1 - x + \sqrt{1 + x^{2}}) + \frac{l n (4 - 2 \sqrt{2})}{2} c = f^{^{'}} (x) \Rightarrow$ $f (x) = \int α l n ∣ \sqrt{2} - 1 - x - \sqrt{1 + x^{2}} ∣) d x + β \int l n ∣ \sqrt{2} - 1 + \sqrt{1 + x^{2}} ∣ d x$ $+ \frac{c x}{2} l n (4 - 2 \sqrt{2}) + C . . . . b e c o n t i n u e d . . .$

Question Number 55296 Answers: 1 Comments: 0

simplify the following expression if x<0 ∣4x−(√((3x−1)^2 ))∣

$s i m p l i f y t h e f o l l o w i n g e x p r e s s i o n i f$ $x < 0$ $∣ 4 x - \sqrt{{(3 x - 1)}^{2}} ∣$

Question Number 55295 Answers: 0 Comments: 0

Question Number 55245 Answers: 1 Comments: 0

Question Number 55240 Answers: 1 Comments: 0

make r the subject of the relation m=((4(√(u+r)))/(v−r))

$m a k e r t h e s u b j e c t o f t h e r e l a t i o n$ $m = \frac{4 \sqrt{u + r}}{v - r}$

Question Number 55237 Answers: 1 Comments: 1

∫_0 ^3 ∫_1 ^2 (x^3 +y^2 )dxdy

$\int_{0}^{3} \int_{1}^{2} (x^{3} + y^{2}) dxdy$

Question Number 55230 Answers: 0 Comments: 1

calculate lim_(ξ→0) ∫_1 ^(1+ξ) ((arctan(ξt))/t) dt .

$c a l c u l a t e {l i m}_{ξ \to 0} \int_{1}^{1 + ξ} \frac{a r c t a n (ξ t)}{t} d t .$

Question Number 55229 Answers: 0 Comments: 1

calculate lim_(n→+∞) ∫_0 ^n (e^(nx) /(1+nx^2 )) dx .

$c a l c u l a t e {l i m}_{n \to + \infty} \int_{0}^{n} \frac{e^{n x}}{1 + {n x}^{2}} d x .$

Question Number 55224 Answers: 3 Comments: 0

log_2 (x^2 +7x−2)=log_2 (x^2 +3x−6)+log_4 8 find x

${l o g}_{2} (x^{2} + 7 x - 2) = {l o g}_{2} (x^{2} + 3 x - 6) + {l o g}_{4} 8$ $f i n d x$

Question Number 55223 Answers: 0 Comments: 5

Question Number 55217 Answers: 1 Comments: 0

Question Number 55214 Answers: 1 Comments: 1

let U_n =∫_(1/n) ^1 (√(x^2 +(3/n)))dx .calculate lim_(n→+∞) U_n

$l e t U_{n} = \int_{\frac{1}{n}}^{1} \sqrt{x^{2} + \frac{3}{n}} d x . c a l c u l a t e {l i m}_{n \to + \infty} U_{n}$

Question Number 55211 Answers: 1 Comments: 0

Question Number 55205 Answers: 0 Comments: 0

Please, can you help me with the following? A boat B is found at 40km at at the east of a boat A. A is moving at 30° east at 20km/h, B is moving qt 10km/h at 30°west. After how many hours will the two boats be the closest to each other? What will therefore be the position of the boat B and the boat A? Thank you

$Please, can you help me with the following ?$ $A boat B is found at 40 km at at the east of$ $a boat A . A is moving at 30 ° east at 20 km / h,$ $B is moving qt 10 km / h at 30 ° west .$ $After how many hours will the two boats$ $be the closest to each other ?$ $What will therefore be the position of the$ $boat B and the boat A ?$ $Thank you$

Question Number 55198 Answers: 2 Comments: 1

x^5 −4x^4 +6x^3 +8x−32=0 Find at least one root.

$x^{5} - 4 x^{4} + 6 x^{3} + 8 x - 32 = 0$ $F i n d a t l e a s t o n e r o o t .$

Question Number 55197 Answers: 0 Comments: 4

∫(dα/(1−sin^3 α))=? ∫(dβ/(1−cos^3 β))=? ∫(dγ/(1−tan^3 γ))=?

$\int \frac{d α}{1 - \sin^{3} α} = ?$ $\int \frac{d β}{1 - \cos^{3} β} = ?$ $\int \frac{d γ}{1 - \tan^{3} γ} = ?$

Question Number 55195 Answers: 1 Comments: 0

x^2 +ax+(√2)b=0,has 2 roots:c and d,also x^2 +cx+(√2)d=0,has 2 roots:a and b.such that:a, b, c, d,are defferent non zero numbers. find possible value(s) for:a^2 +b^2 +c^2 +d^2 .

$x^{2} + ax + \sqrt{2} b = 0, has 2 roots : c and d, also$ $x^{2} + cx + \sqrt{2} d = 0, has 2 roots : a and b . such$ $that : a, b, c, d, are defferent non zero$ $numbers .$ $find possible value (s) for : a^{2} + b^{2} + c^{2} + d^{2} .$

Question Number 55193 Answers: 1 Comments: 0

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