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DifferentiationQuestion and Answers: Page 11

Question Number 164339 Answers: 1 Comments: 0

In AB^Δ C : cos^( 2) (A )+ cos^( 2) (B )+ cos^( 2) ( C )=1 . Prove that AB^Δ C is right angled. −−−−−−−−

$I n A \overset{Δ}{B C} : {c o s}^{2} (A) + {c o s}^{2} (B) + {c o s}^{2} (C) = 1 .$ $P r o v e t h a t A \overset{Δ}{B C} i s r i g h t a n g l e d .$ $- - - - - - - -$

Question Number 164103 Answers: 1 Comments: 0

prove that Ω=∫_0 ^( 1) ln(((1+x)/(1−x)) ).(dx/(x (√( 1−x^( 2) )))) = (π^( 2) /2) −− m.n−−

$p r o v e t h a t$ $Ω = \int_{0}^{1} \ln (\frac{1 + x}{1 - x}) . \frac{d x}{x \sqrt{1 - x^{2}}} = \frac{π^{2}}{2}$ $- - m . n - -$

Question Number 164059 Answers: 0 Comments: 0

Air leaks from a spherical ballon so that it maintains its shape at a rate of 25 cc/m .What is the rate of change in the length of the radius of the balloon when the radius is 5 cm

$Air leaks from a spherical ballon so that$ $it maintains its shape at a rate of 25 cc / m$ $. What is the rate of change in the length$ $of the radius of the balloon when the radius$ $is 5 cm$

Question Number 163928 Answers: 1 Comments: 0

f(x)=((2x^(100!) )/(100!))+x^(100) +1 find ((d^(100!) f(x))/dx^(100!) )=?

$f (x) = \frac{2 x^{100!}}{100!} + x^{100} + 1$ $f i n d \frac{d^{100!} f (x)}{{d x}^{100!}} = ?$

Question Number 163732 Answers: 0 Comments: 0

If , 𝛗 = ∫_(−∞) ^( +∞) (( sin(x).ln^( 2) (x ))/x) then find : Im (𝛗 ) = ? ■ m.n

$I f, ϕ = \int_{- \infty}^{+ \infty} \frac{s i n (x) . {l n}^{2} (x)}{x} t h e n$ $f i n d : I m (ϕ) = ? ◼ m . n$

Question Number 163704 Answers: 2 Comments: 0

Ω= Σ_(n=1) ^∞ n(ζ (1+n) −1) =?

$Ω = \underset{n = 1}{\sum^{\infty}} n (ζ (1 + n) - 1) = ?$

Question Number 163700 Answers: 3 Comments: 0

K(x) = ((3 cos x)/(5+4sin x)) {: ((max K(x))),((min K(x))) } =?

$K (x) = \frac{3 \cos x}{5 + 4 \sin x}$ $\begin{matrix} m a x K (x) \\ m i n K (x) \end{matrix}} = ?$

Question Number 163682 Answers: 1 Comments: 0

Ω= ∫_0 ^( 1) (Li_( 2) ^ (x ))^( 2) dx = ? ■ m.n −−− −−−

$Ω = \int_{0}^{1} {({Li}_{2}^{} (x))}^{2} d x = ? ◼ m . n$ $- - - - - -$

Question Number 163613 Answers: 0 Comments: 0

Question Number 163533 Answers: 0 Comments: 5

Re^ soudre (∂^2 u/∂x^2 )+(∂^2 u/∂y^2 )=e^(2x+y)

$R \overset{´}{e s o u d r e} \frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}} = e^{2 x + y}$

Question Number 163510 Answers: 0 Comments: 1

If 𝛗=∫_0 ^( (π/2)) (( 1)/( (√(sin^( 5) (x).cos(x))) +(√(cos^( 5) (x).sin(x)))))dx = find the value of : Γ^( 2) ((3/4) ). 𝛗

$I f$ $ϕ = \int_{0}^{\frac{π}{2}} \frac{1}{\sqrt{{s i n}^{5} (x) . c o s (x)} + \sqrt{{c o s}^{5} (x) . s i n (x)}} d x =$ $f i n d t h e v a l u e o f : Γ^{2} (\frac{3}{4}) . ϕ$

Question Number 163487 Answers: 1 Comments: 0

Ω= ∫_0 ^( 1) (( sin^( 2) ( ln(x )). ln (x))/( (√x))) dx=? −−−−−

$Ω = \int_{0}^{1} \frac{{s i n}^{2} (\ln (x)) . \ln (x)}{\sqrt{x}} d x = ?$ $- - - - -$

Question Number 163400 Answers: 2 Comments: 0

prove Ω= ∫_0 ^( ∞) cot^( −1) (1+x^( 2) )=((√((1/( (√2)))−(1/2))) ) π

$p r o v e$ $Ω = \int_{0}^{\infty} {c o t}^{- 1} (1 + x^{2}) = (\sqrt{\frac{1}{\sqrt{2}} - \frac{1}{2}}) π$

Question Number 163367 Answers: 2 Comments: 0

Question Number 163271 Answers: 0 Comments: 0

put : gcd( a , b )= (a, b ) if ( a ,b )= (a ,c )= (b ,c )=1 prove that : (abc , ab +ac +bc )=1

$p u t : g c d (a, b) = (a, b)$ $i f (a, b) = (a, c) = (b, c) = 1$ $p r o v e t h a t : (a b c, a b + a c + b c) = 1$

Question Number 163257 Answers: 0 Comments: 0

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

$R e (\int_{0}^{1} {s i n}^{- 1} (\frac{1}{1 - x^{2}}) d x) = ?$

Question Number 163191 Answers: 0 Comments: 1

Question Number 163134 Answers: 1 Comments: 0

prove or disprove ∫_(2π) ^( 4π) (( sin(x))/x) dx >0 because ∫_(2π) ^( 3π) (( sin(x ))/x) dx > ∫_(3π) ^( 4π) ((∣sin(x)∣)/x) dx

$p r o v e o r d i s p r o v e$ $\int_{2 π}^{4 π} \frac{s i n (x)}{x} d x > 0$ $b e c a u s e$ $\int_{2 π}^{3 π} \frac{s i n (x)}{x} d x > \int_{3 π}^{4 π} \frac{∣ s i n (x) ∣}{x} d x$

Question Number 163080 Answers: 1 Comments: 0

F(x)= (((x^2 −4x)^2 ))^(1/3) {: ((local maximum)),((absolut maximum)) } =?

$F (x) = \sqrt[3]{{(x^{2} - 4 x)}^{2}}$ $\begin{matrix} l o c a l m a x i m u m \\ a b s o l u t m a x i m u m \end{matrix}} = ?$

Question Number 163045 Answers: 1 Comments: 0

prove that Σ_(n=1) ^∞ (( ( 2n+1 )!!)/((2n )!!)) (1/(2^( n) (2n +1)^( 2) )) =((π(√2))/4)−1

$p r o v e t h a t$ $\underset{n = 1}{\sum^{\infty}} \frac{(2 n + 1)!!}{(2 n)!!} \frac{1}{2^{n} {(2 n + 1)}^{2}} = \frac{π \sqrt{2}}{4} - 1$

Question Number 163033 Answers: 1 Comments: 0

Ω= ∫_0 ^( ∞) (( x − sin (x ))/x^( 3) )dx −−− solution−−− Ω=^(I.B.P) [ ((−1)/(2 x^( 2) )) (x−sin(x))]_0 ^∞ +(1/2) ∫_0 ^( ∞) ((1−cos (x))/x^( 2) )dx = (1/2) ∫_0 ^( ∞) (( 2sin^( 2) ((x/2)))/x^( 2) )dx=∫_0 ^( ∞) ((sin^( 2) ((x/2)))/x^( 2) )dx =^((x/2) = α) (1/2)∫_0 ^( ∞) ((sin^( 2) ( α))/α^( 2) ) dα = (1/2) [((−1)/α) sin^( 2) (α)]_0 ^∞ +(1/2)∫_0 ^( ∞) ((sin(2α))/α)dα =^(2α=ϕ) (1/2) ∫_0 ^( ∞) (( sin(ϕ ))/ϕ) dϕ =(π/4) −− Ω= (π/4) −−−

$Ω = \int_{0}^{\infty} \frac{x - s i n (x)}{x^{3}} d x$ $- - - s o l u t i o n - - -$ $Ω \overset{I . B . P}{=} {[\frac{- 1}{2 x^{2}} (x - s i n (x))]}_{0}^{\infty} + \frac{1}{2} \int_{0}^{\infty} \frac{1 - c o s (x)}{x^{2}} d x$ $= \frac{1}{2} \int_{0}^{\infty} \frac{2 {s i n}^{2} (\frac{x}{2})}{x^{2}} d x = \int_{0}^{\infty} \frac{{s i n}^{2} (\frac{x}{2})}{x^{2}} d x$ $\overset{\frac{x}{2} = α}{=} \frac{1}{2} \int_{0}^{\infty} \frac{{s i n}^{2} (α)}{α^{2}} d α$ $= \frac{1}{2} {[\frac{- 1}{α} {s i n}^{2} (α)]}_{0}^{\infty} + \frac{1}{2} \int_{0}^{\infty} \frac{s i n (2 α)}{α} d α$ $\overset{2 α = φ}{=} \frac{1}{2} \int_{0}^{\infty} \frac{s i n (φ)}{φ} d φ = \frac{π}{4}$ $- - Ω = \frac{π}{4} - - -$

Question Number 163002 Answers: 0 Comments: 0

prove that i:Σ_(n=0) ^∞ (((−1 )^( n) )/((n +(1/2))cosh(n+(1/2))π)) =(π/4) ii: ∫_0 ^( 1) (( sin( π x ))/(x^( x) ( 1−x )^( 1−x) )) (dx/(1+x)) =(π/4) −−−

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

Question Number 162939 Answers: 0 Comments: 0

lim_( x→ 3) ( a ⌊x ⌋ + ⌊ −x⌋).tan(((πx)/2) )=−∞ a ∈ ?

${l i m}_{x \to 3} (a ⌊ x ⌋ + ⌊ - x ⌋) . t a n (\frac{π x}{2}) = - \infty$ $a \in ?$

Question Number 162924 Answers: 2 Comments: 0

𝛗 =∫_0 ^( ∞) (( e^( −x^( 2) ) .ln( x ))/( (√x))) dx=λ Γ((1/4)) λ=? ■

$ϕ = \int_{0}^{\infty} \frac{e^{- x^{2}} . \ln (x)}{\sqrt{x}} d x = λ Γ (\frac{1}{4})$ $λ = ? ◼$

Question Number 162804 Answers: 2 Comments: 0

Ω = ∫ sin^( 2) (x).cos^( 4) (x ) dx

$Ω = \int {s i n}^{2} (x) . {c o s}^{4} (x) d x$

Question Number 162701 Answers: 1 Comments: 0

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