Question and Answers Forum

All Questions Topic List

DifferentiationQuestion and Answers: Page 13

Question Number 160752 Answers: 1 Comments: 0

solve (d^2 y/dx^2 )−y=x^2 sin3x

$s o l v e$ $\frac{d^{2} y}{{d x}^{2}} - y = x^{2} s i n 3 x$

Question Number 160603 Answers: 1 Comments: 0

(x+(√(x^2 +1)))(y+(√(y^2 +1)))=2021 ∀x,y∈R^+ . min (x+y)=?

$(x + \sqrt{x^{2} + 1}) (y + \sqrt{y^{2} + 1}) = 2021$ $\forall x, y \in R^{+} . \min (x + y) = ?$

Question Number 160577 Answers: 1 Comments: 0

Ω=∫_0 ^( 1) tan^( −1) (x).ln(x) = ? −−−−solution−−−− f(a)=∫_0 ^( 1) tan^( −1) ( x) .x^( a) dx = Σ_(n=1) ^∞ (( (−1)^( n−1) )/(2n−1)) ∫_0 ^( 1) x^( 2n+a−1) dx = Σ_(n=0) ^∞ (((−1)^( n−1) )/(2n−1)) ((1/(2n+ a)) ) Ω= f ′ (a )∣_(a=0) = Σ_(n=1) ^∞ (((−1)^n )/((2n−1)( 2n+ a)^( 2) )) Ω= f ′ (0 )=(1/4)Σ(( (−1 )^(n−1) (2n−1−2n ) )/(( 2n−1)n^( 2) )) =(1/4) Σ_(n=1) ^∞ (((−1)^(n−1) )/n^( 2) ) − Σ_(n=1) ^∞ (((−1)^( n−1) )/((2n−1)(2n))) = (π^( 2) /(48)) −{ Σ_(n=1) ^∞ {(((−1)^(n−1) )/(2n−1)) −(((−1)^( n−1) )/(2n))}} ∴ Ω = (π^( 2) /(48)) − (π/4) +(1/2) ln(2)

$Ω = \int_{0}^{1} {t a n}^{- 1} (x) . l n (x) = ?$ $- - - - s o l u t i o n - - - -$ $f (a) = \int_{0}^{1} {t a n}^{- 1} (x) . x^{a} d x$ $= \underset{n = 1}{\sum^{\infty}} \frac{{(- 1)}^{n - 1}}{2 n - 1} \int_{0}^{1} x^{2 n + a - 1} d x$ $= \underset{n = 0}{\sum^{\infty}} \frac{{(- 1)}^{n - 1}}{2 n - 1} (\frac{1}{2 n + a})$ $Ω = f^{'} (a) ∣_{a = 0} = \underset{n = 1}{\sum^{\infty}} \frac{{(- 1)}^{n}}{(2 n - 1) {(2 n + a)}^{2}}$ $Ω = f^{'} (0) = \frac{1}{4} Σ \frac{{(- 1)}^{n - 1} (2 n - 1 - 2 n)}{(2 n - 1) n^{2}}$ $= \frac{1}{4} \underset{n = 1}{\sum^{\infty}} \frac{{(- 1)}^{n - 1}}{n^{2}} - \underset{n = 1}{\sum^{\infty}} \frac{{(- 1)}^{n - 1}}{(2 n - 1) (2 n)}$ $= \frac{π^{2}}{48} - {\underset{n = 1}{\sum^{\infty}} {\frac{{(- 1)}^{n - 1}}{2 n - 1} - \frac{{(- 1)}^{n - 1}}{2 n}}}$ $∴ Ω = \frac{π^{2}}{48} - \frac{π}{4} + \frac{1}{2} l n (2)$

Question Number 160558 Answers: 1 Comments: 0

solve ⌊ x− (√(1−x^( 2) )) ⌋+⌊ x+ (√(1−x^( 2) )) ⌋=0

$s o l v e$ $⌊ x - \sqrt{1 - x^{2}} ⌋ + ⌊ x + \sqrt{1 - x^{2}} ⌋ = 0$

Question Number 160545 Answers: 0 Comments: 0

Σ_(k=1) ^n (n/(n^2 +k)) (1/n)Σ_(k=1 ) ^n cos((1/( (√(n+k))))) convergente?

$\underset{k = 1}{\sum^{n}} \frac{n}{n^{2} + k}$ $\frac{1}{n} \underset{k = 1}{\sum^{n}} c o s (\frac{1}{\sqrt{n + k}})$ $c o n v e r g e n t e ?$

Question Number 160530 Answers: 2 Comments: 0

Question Number 160528 Answers: 0 Comments: 0

(2cosh(x)cos(y))dx+(sinh(x)sin(y))dy=0

$(2 \cosh (x) \cos (y)) dx + (\sinh (x) \sin (y)) dy = 0$

Question Number 160363 Answers: 0 Comments: 0

s>0 limΣ_(k=1) ^(2n) ( −1)^( k) .((( k)/(2n)) )^( s) = ?

$s > 0$ $l i m \underset{k = 1}{\sum^{2 n}} {(- 1)}^{k} . {(\frac{k}{2 n})}^{s} = ?$

Question Number 160252 Answers: 2 Comments: 0

∫_0 ^( 1) (( ln^( 2) (1−x )ln(x))/x)dx=?

$\int_{0}^{1} \frac{{l n}^{2} (1 - x) l n (x)}{x} d x = ?$

Question Number 160223 Answers: 0 Comments: 0

Ω:=∫_0 ^( ∞) (( x)/(( e^( x) +e^( −x) )^( 3) )) dx =?

$Ω := \int_{0}^{\infty} \frac{x}{{(e^{x} + e^{- x})}^{3}} d x = ?$

Question Number 160159 Answers: 0 Comments: 0

prove that .. ∫_0 ^( ∞) (( x)/(cosh^( 3) (x ))) dx = G − (1/2) ■ G: catalan constant

$p r o v e t h a t . .$ $\int_{0}^{\infty} \frac{x}{{c o s h}^{3} (x)} d x = G - \frac{1}{2} ◼$ $G : c a t a l a n c o n s t a n t$

Question Number 159966 Answers: 0 Comments: 2

a y=(√(x+(√(x+(√(x+.....)))))) b y=(√(x(√(x(√(x(√(x.....)))))))) find (dy/dx)

$a y = \sqrt{x + \sqrt{x + \sqrt{x + . . . . .}}}$ $b y = \sqrt{x \sqrt{x \sqrt{x \sqrt{x . . . . .}}}}$ $f i n d \frac{d y}{d x}$

Question Number 159918 Answers: 0 Comments: 2

y = sin 8x cos 4x y^((n)) =?

$y = \sin 8 x \cos 4 x$ $y^{(n)} = ?$

Question Number 159874 Answers: 0 Comments: 3

Given the curve y=x^4 +3x^3 −6x^2 −3x determine for which value of α the tangent to the curve from point P(α,0) is maximum.

$G i v e n t h e c u r v e y = x^{4} + 3 x^{3} - 6 x^{2} - 3 x$ $d e t e r m i n e f o r w h i c h v a l u e$ $o f α t h e t a n g e n t t o t h e c u r v e$ $f r o m p o i n t P (α, 0) i s m a x i m u m .$

Question Number 159982 Answers: 1 Comments: 0

(1+bf(x))f′′(x)=(p/(λa)) solve this equation: find f(x)

$(1 + b f (x)) f^{″} (x) = \frac{p}{λ a}$ $s o l v e t h i s e q u a t i o n : f i n d f (x)$

Question Number 159854 Answers: 1 Comments: 0

Ω := ∫_0 ^( (π/4)) x.ln(sin(x))dx= ?

$Ω := \int_{0}^{\frac{π}{4}} x . l n (s i n (x)) d x = ?$

Question Number 159664 Answers: 0 Comments: 2

prove that : Φ = ∫_0 ^( ∞) (( sin^( 4) (x))/x^( 3) )dx= ln(2) −−−−−−−−−

$p r o v e t h a t :$ $Φ = \int_{0}^{\infty} \frac{{s i n}^{4} (x)}{x^{3}} d x = l n (2)$ $- - - - - - - - -$

Question Number 159648 Answers: 2 Comments: 0

y = sin^2 (2x) y^((n)) =?

$y = \sin^{2} (2 x)$ $y^{(n)} = ?$

Question Number 159592 Answers: 1 Comments: 0

calculate: I :=∫_0 ^( ∞) (((arctan(x))/x))^3 dx=?

$c a l c u l a t e :$ $I := \int_{0}^{\infty} {(\frac{a r c t a n (x)}{x})}^{3} d x = ?$

Question Number 159540 Answers: 1 Comments: 0

prove that : 𝛗 := ∫_0 ^( ∞) (( sin((√x) ).sin((π/3) +(√x) ).sin(((2π)/3)+(√x) ).ln((1/x^( 2) ) ))/x)dx=^? π.(γ + ln(3) ) −−−−−−−−−− m.n

$p r o v e t h a t :$ $ϕ := \int_{0}^{\infty} \frac{s i n (\sqrt{x}) . s i n (\frac{π}{3} + \sqrt{x}) . s i n (\frac{2 π}{3} + \sqrt{x}) . l n (\frac{1}{x^{2}})}{x} d x \overset{?}{=} π . (γ + l n (3))$ $- - - - - - - - - - m . n$

Question Number 159526 Answers: 0 Comments: 0

Ω= ∫_0 ^( ∞) (( sin^( 3) (x)ln(x))/x) dx=^(??) (π/8) (ln(3)−2γ) −−−−− solution.. Ω=∫_0^ ^( ∞) {(((3/4) sin(x)−(1/4) sin(3x))/x)} ln(x)dx = (3/4) (((−πγ)/2))− (1/4){ ∫_0 ^( ∞) ((sin(3x)ln(x))/x)dx=Ψ} ∴ Ψ := ∫_0 ^( ∞) (( sin(x).[ln(x)−ln(3)])/x)dx := −((πγ)/2) − ((ln(3).π)/2) ∴ Ω := ((−3πγ)/8) +((πγ)/8) +((π.ln(3))/8) := ((−2πγ)/8) + ((π.ln(3))/8) = (π/8) ( ln(3)−2γ )

$Ω = \int_{0}^{\infty} \frac{{s i n}^{3} (x) l n (x)}{x} d x \overset{? ?}{=} \frac{π}{8} (l n (3) - 2 γ)$ $- - - - -$ $s o l u t i o n . .$ $Ω = \int_{0^{}}^{\infty} {\frac{\frac{3}{4} s i n (x) - \frac{1}{4} s i n (3 x)}{x}} l n (x) d x$ $= \frac{3}{4} (\frac{- π γ}{2}) - \frac{1}{4} {\int_{0}^{\infty} \frac{s i n (3 x) l n (x)}{x} d x = Ψ}$ $∴ Ψ := \int_{0}^{\infty} \frac{s i n (x) . [l n (x) - l n (3)]}{x} d x$ $:= - \frac{π γ}{2} - \frac{l n (3) . π}{2}$ $∴ Ω := \frac{- 3 π γ}{8} + \frac{π γ}{8} + \frac{π . l n (3)}{8}$ $:= \frac{- 2 π γ}{8} + \frac{π . l n (3)}{8} = \frac{π}{8} (l n (3) - 2 γ)$

Question Number 159474 Answers: 1 Comments: 0

nice integral. prove that ∫^( ∞) _0 (( tan^( −1) (2x)+ tan^( −1) ((x/2) ))/(1+x^( 2) ))dx=(π^( 2) /4)

$n i c e i n t e g r a l .$ $p r o v e t h a t$ ${\int_{0}}^{\infty} \frac{{t a n}^{- 1} (2 x) + {t a n}^{- 1} (\frac{x}{2})}{1 + x^{2}} d x = \frac{π^{2}}{4}$

Question Number 159465 Answers: 1 Comments: 0

simplify ξ := Σ_(n=1) ^∞ ( (( 1)/(Σ_(k=1) ^n k^3 )) )=?

$s i m p l i f y$ $ξ := \underset{n = 1}{\sum^{\infty}} (\frac{1}{\underset{k = 1}{\sum^{n}} k^{3}}) = ?$

Question Number 159447 Answers: 1 Comments: 1

#calculate# Ω := ∫_0 ^( 1) ∫_0 ^( 1) (( x^( (t/2)) −x^( t) )/(1 − x)) dx dt = ? −−−m.n−−−

$You can't use 'macro parameter character #' in math mode$ $Ω := \int_{0}^{1} \int_{0}^{1} \frac{x^{\frac{t}{2}} - x^{t}}{1 - x} d x d t = ?$ $- - - m . n - - -$

Question Number 159021 Answers: 1 Comments: 0

Question Number 158803 Answers: 1 Comments: 1

Pg 8 Pg 9 Pg 10 Pg 11 Pg 12 Pg 13 Pg 14 Pg 15 Pg 16 Pg 17

Contact: info@tinkutara.com