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Question Number 43170 Answers: 1 Comments: 3

let A_n = ∫_0 ^∞ [n e^(−x) ]dx with n integr natural. 1) calculate A_n 2) find lim_(n→+∞) A_n .

$${let}\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{\infty} \:\:\left[{n}\:{e}^{−{x}} \right]{dx}\:\:{with}\:{n}\:{integr}\:{natural}. \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{A}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} . \\ $$

Question Number 43147 Answers: 0 Comments: 2

If ∫_0 ^∞ e^(−x^2 ) dx = ((√π)/(2 )) , then prove that ∫_0 ^∞ e^(−ax^2 ) dx = (√(π/(4a))) where a>0.

$$\mathrm{If}\:\:\:\int_{\mathrm{0}} ^{\infty} \:\mathrm{e}^{−{x}^{\mathrm{2}} } {dx}\:=\:\frac{\sqrt{\pi}}{\mathrm{2}\:}\:, \\ $$$$\mathrm{then}\:{prove}\:{that}\:\int_{\mathrm{0}} ^{\infty} \mathrm{e}^{−\mathrm{a}{x}^{\mathrm{2}} } {dx}\:=\:\sqrt{\frac{\pi}{\mathrm{4a}}} \\ $$$$\mathrm{where}\:\mathrm{a}>\mathrm{0}. \\ $$

Question Number 43145 Answers: 1 Comments: 4

∫_0 ^∞ [ 2e^(−x) ]dx = ? where [.]= gif.

$$\int_{\mathrm{0}} ^{\infty} \:\left[\:\mathrm{2e}^{−{x}} \right]{dx}\:=\:?\: \\ $$$${where}\:\left[.\right]=\:{gif}. \\ $$

Question Number 43135 Answers: 1 Comments: 4

Construct a triangle ΔABC with ∠B=50° AC=6 cm AB+BC=8 cm see also Q42942.

$${Construct}\:{a}\:{triangle}\:\Delta{ABC}\:{with} \\ $$$$\angle{B}=\mathrm{50}° \\ $$$${AC}=\mathrm{6}\:{cm} \\ $$$${AB}+{BC}=\mathrm{8}\:{cm} \\ $$$$ \\ $$$${see}\:{also}\:{Q}\mathrm{42942}. \\ $$

Question Number 43131 Answers: 2 Comments: 1

Question Number 43125 Answers: 1 Comments: 1

Question Number 43100 Answers: 1 Comments: 2

let f(x) = ∫_0 ^(π/2) ((cosθ)/(1+xsinθ))dθ 1) determine a explicit form of f(x) 2) calculate ∫_0 ^(π/2) ((sin(2θ))/((1+xsinθ)^2 ))dθ 3) find the values of ∫_0 ^(π/2) ((cosθ)/(1+2cosθ))dθ and ∫_0 ^(π/2) ((sin(2θ))/((1+3sinθ)^2 ))dθ .

$${let}\:{f}\left({x}\right)\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\:\:\frac{{cos}\theta}{\mathrm{1}+{xsin}\theta}{d}\theta \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{sin}\left(\mathrm{2}\theta\right)}{\left(\mathrm{1}+{xsin}\theta\right)^{\mathrm{2}} }{d}\theta \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{values}\:{of}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{cos}\theta}{\mathrm{1}+\mathrm{2}{cos}\theta}{d}\theta\:\:\:{and}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\:\:\frac{{sin}\left(\mathrm{2}\theta\right)}{\left(\mathrm{1}+\mathrm{3}{sin}\theta\right)^{\mathrm{2}} }{d}\theta\:. \\ $$

Question Number 43080 Answers: 0 Comments: 4

Question Number 43072 Answers: 1 Comments: 0

Question Number 43064 Answers: 0 Comments: 8

Question Number 43058 Answers: 0 Comments: 1

calculate ∫_0 ^∞ ((x sin(((πx)/2)))/({1+(x+1)^2 }{1+(x−1)^2 }))dx

$${calculate}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{x}\:{sin}\left(\frac{\pi{x}}{\mathrm{2}}\right)}{\left\{\mathrm{1}+\left({x}+\mathrm{1}\right)^{\mathrm{2}} \right\}\left\{\mathrm{1}+\left({x}−\mathrm{1}\right)^{\mathrm{2}} \right\}}{dx} \\ $$

Question Number 43057 Answers: 0 Comments: 1

find the value of I = ∫_0 ^∞ ((cos(πx^2 ) −sin(πx^2 ))/((1+x^2 )^2 ))dx

$${find}\:{the}\:{value}\:{of}\:\:\:{I}\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{cos}\left(\pi{x}^{\mathrm{2}} \right)\:−{sin}\left(\pi{x}^{\mathrm{2}} \right)}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx} \\ $$

Question Number 43092 Answers: 0 Comments: 5

Question Number 43128 Answers: 1 Comments: 5

Question Number 43036 Answers: 1 Comments: 1

Question Number 43031 Answers: 2 Comments: 2

Question Number 43027 Answers: 2 Comments: 1

Question Number 43023 Answers: 1 Comments: 1

Question Number 43011 Answers: 0 Comments: 2

Question Number 43008 Answers: 1 Comments: 0

(y′)^2 =−1+sin x y=?

$$\left({y}'\right)^{\mathrm{2}} =−\mathrm{1}+\mathrm{sin}\:{x} \\ $$$${y}=? \\ $$

Question Number 43007 Answers: 1 Comments: 1

calculate A_n =Σ_(k=0) ^n k C_n ^k B_n = Σ_(k=0) ^n k^2 C_n ^k C_n =Σ_(k=0) ^n k^3 C_n ^k

$${calculate}\: \\ $$$${A}_{{n}} =\sum_{{k}=\mathrm{0}} ^{{n}} \:{k}\:{C}_{{n}} ^{{k}} \\ $$$${B}_{{n}} =\:\sum_{{k}=\mathrm{0}} ^{{n}} \:{k}^{\mathrm{2}} \:{C}_{{n}} ^{{k}} \\ $$$${C}_{{n}} =\sum_{{k}=\mathrm{0}} ^{{n}} \:{k}^{\mathrm{3}} \:{C}_{{n}} ^{{k}} \\ $$

Question Number 43005 Answers: 0 Comments: 19

The base of triangle passes through a fixed point p(a,b) and its sides are respectively bisected at right angles by the line x+y=0 and x=9y if the locus of the third vartex is a circle. then find its equation.

$$\mathrm{The}\:\mathrm{base}\:\mathrm{of}\:\mathrm{triangle}\:\mathrm{passes}\:\mathrm{through} \\ $$$$\mathrm{a}\:\mathrm{fixed}\:\mathrm{point}\:\mathrm{p}\left(\mathrm{a},\mathrm{b}\right)\:\mathrm{and}\:\mathrm{its}\:\mathrm{sides} \\ $$$$\mathrm{are}\:\mathrm{respectively}\:\mathrm{bisected}\:\mathrm{at}\:\mathrm{right}\:\mathrm{angles}\: \\ $$$$\mathrm{by}\:\mathrm{the}\:\mathrm{line}\:\mathrm{x}+\mathrm{y}=\mathrm{0}\:\mathrm{and}\:\mathrm{x}=\mathrm{9y} \\ $$$$\mathrm{if}\:\mathrm{the}\:\mathrm{locus}\:\mathrm{of}\:\mathrm{the}\:\mathrm{third}\:\mathrm{vartex}\:\mathrm{is}\:\mathrm{a}\:\mathrm{circle}. \\ $$$$\mathrm{then}\:\mathrm{find}\:\mathrm{its}\:\mathrm{equation}. \\ $$

Question Number 43003 Answers: 1 Comments: 0

let u_n = Σ_(1≤i<j≤n) (1/(√(ij))) 1) find a equivalent of u_n 2)calculate lim_(n→+∞) u_n

$${let}\:{u}_{{n}} =\:\sum_{\mathrm{1}\leqslant{i}<{j}\leqslant{n}} \:\:\frac{\mathrm{1}}{\sqrt{{ij}}} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{equivalent}\:{of}\:{u}_{{n}} \\ $$$$\left.\mathrm{2}\right){calculate}\:{lim}_{{n}\rightarrow+\infty} \:{u}_{{n}} \\ $$

Question Number 43001 Answers: 1 Comments: 0

prove that Π_(k=1) ^n (1+(1/k))>1+Σ_(k=1) ^n (1/k)

$${prove}\:{that}\:\prod_{{k}=\mathrm{1}} ^{{n}} \left(\mathrm{1}+\frac{\mathrm{1}}{{k}}\right)>\mathrm{1}+\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{k}} \\ $$

Question Number 43000 Answers: 2 Comments: 0

prove that (√(2+(√(2+....+(√2))))) =2cos((π/2^n ))

$${prove}\:{that}\:\sqrt{\mathrm{2}+\sqrt{\mathrm{2}+....+\sqrt{\mathrm{2}}}}\:\:=\mathrm{2}{cos}\left(\frac{\pi}{\mathrm{2}^{{n}} }\right) \\ $$

Question Number 42999 Answers: 0 Comments: 1

let f(x)=(√x)+(1/(x−1)) 1) calculate f^((n)) (2) 2) if f(x) =Σ_(n=0) ^∞ a_n (x−2)^n find the sequence a_n

$${let}\:{f}\left({x}\right)=\sqrt{{x}}+\frac{\mathrm{1}}{{x}−\mathrm{1}} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{\left({n}\right)} \left(\mathrm{2}\right) \\ $$$$\left.\mathrm{2}\right)\:{if}\:{f}\left({x}\right)\:=\sum_{{n}=\mathrm{0}} ^{\infty} \:{a}_{{n}} \left({x}−\mathrm{2}\right)^{{n}} \:{find}\:{the} \\ $$$${sequence}\:{a}_{{n}} \\ $$

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