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let put F(x)= ∫_0 ^∞ e^(−tx) ((sint)/t) dt with x≥0 we accept that F is class C^1 on [0,∝[ calculate (∂F/∂x) and find F(x) then find the value of ∫_0 ^∞ ((sint)/t) dt

$${let}\:{put}\:{F}\left({x}\right)=\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{tx}} \:\frac{{sint}}{{t}}\:{dt}\:\:\:{with}\:\:{x}\geqslant\mathrm{0} \\ $$$${we}\:{accept}\:{that}\:{F}\:{is}\:{class}\:{C}^{\mathrm{1}} \:{on}\:\left[\mathrm{0},\propto\left[\right.\right. \\ $$$${calculate}\:\:\frac{\partial{F}}{\partial{x}}\:\:{and}\:{find}\:{F}\left({x}\right) \\ $$$${then}\:\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\frac{{sint}}{{t}}\:{dt} \\ $$

Question Number 26558 Answers: 0 Comments: 1

find the value of ∫_0 ^∞ ((sinx)/(x(1+x^2 )))dx

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{sinx}}{{x}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{dx} \\ $$

Question Number 26557 Answers: 0 Comments: 0

Question Number 26555 Answers: 0 Comments: 1

(2) Find the 10th trem in the expansion of (2x−(y/2))

$$\left(\mathrm{2}\right)\:\:\boldsymbol{\mathrm{Find}}\:\boldsymbol{\mathrm{the}}\:\mathrm{10}\boldsymbol{\mathrm{th}}\:\boldsymbol{\mathrm{trem}}\:\boldsymbol{\mathrm{in}}\:\boldsymbol{\mathrm{the}} \\ $$$$\boldsymbol{\mathrm{expansion}}\:\boldsymbol{\mathrm{of}}\:\left(\mathrm{2}\boldsymbol{\mathrm{x}}−\frac{\boldsymbol{\mathrm{y}}}{\mathrm{2}}\right) \\ $$

Question Number 26554 Answers: 1 Comments: 0

(2) Find the middle trem(s) in the expansion of following− (x^2 +(1/x^3 ))^(14)

$$\left(\mathrm{2}\right)\:\boldsymbol{\mathrm{Find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{middle}}\:\boldsymbol{\mathrm{trem}}\left(\boldsymbol{\mathrm{s}}\right)\:\boldsymbol{\mathrm{in}}\:\boldsymbol{\mathrm{the}} \\ $$$$\boldsymbol{\mathrm{expansion}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{following}}− \\ $$$$\:\:\:\:\:\left(\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\frac{\mathrm{1}}{\boldsymbol{\mathrm{x}}^{\mathrm{3}} }\right)^{\mathrm{14}} \\ $$

Question Number 26544 Answers: 1 Comments: 0

∫1/x^2 +y^2 dydx

$$\int\mathrm{1}/{x}^{\mathrm{2}} +{y}^{\mathrm{2}} {dydx} \\ $$

Question Number 26534 Answers: 0 Comments: 2

Find the lim_(θ→0) ((3^(sin θ) −1)/θ)

$$\mathrm{Find}\:\:\mathrm{the} \\ $$$$\underset{\theta\rightarrow\mathrm{0}} {\mathrm{lim}}\:\:\frac{\mathrm{3}^{\mathrm{sin}\:\theta} −\mathrm{1}}{\theta} \\ $$

Question Number 26601 Answers: 1 Comments: 0

Use polar co-ordinates to evaluate ∫∫_R e^(−(x^2 +y^2 )) dA, where the region R is enclosed by the circle x^2 +y^2 =1.

$${Use}\:{polar}\:{co}-{ordinates}\:{to}\:{evaluate}\:\int\underset{{R}} {\int}{e}^{−\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)} {dA},\:{where}\:{the}\:{region}\:{R}\:{is}\:{enclosed}\:{by}\:{the}\:{circle}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} =\mathrm{1}. \\ $$

Question Number 26521 Answers: 1 Comments: 1

If (1/1^2 ) + (1/2^2 ) + (1/3^2 ) + ...to ∞ = (π^2 /6), then (1/1^2 ) + (1/3^2 ) + (1/5^2 ) + ... equals

$$\mathrm{If}\:\:\frac{\mathrm{1}}{\mathrm{1}^{\mathrm{2}} }\:+\:\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{2}} }\:+\:\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{2}} }\:+\:...\mathrm{to}\:\infty\:=\:\frac{\pi^{\mathrm{2}} }{\mathrm{6}},\:\mathrm{then} \\ $$$$\frac{\mathrm{1}}{\mathrm{1}^{\mathrm{2}} }\:+\:\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{2}} }\:+\:\frac{\mathrm{1}}{\mathrm{5}^{\mathrm{2}} }\:+\:...\:\mathrm{equals} \\ $$

Question Number 26877 Answers: 1 Comments: 2

find the value of Σ_(n=1) ^∝ (1/(n(n+1)(n+2)(n+3))) .

$${find}\:{the}\:{value}\:{of}\:\:\sum_{{n}=\mathrm{1}} ^{\propto} \:\frac{\mathrm{1}}{{n}\left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)\left({n}+\mathrm{3}\right)}\:\:. \\ $$

Question Number 26851 Answers: 0 Comments: 1

Two integers x and y are chosen with replacement out of the set {0, 1, 2, 3,..., 10}. Then the probability that ∣x−y∣>5 is

$$\mathrm{Two}\:\mathrm{integers}\:{x}\:\mathrm{and}\:{y}\:\mathrm{are}\:\mathrm{chosen}\:\mathrm{with} \\ $$$$\mathrm{replacement}\:\mathrm{out}\:\mathrm{of}\:\mathrm{the}\:\mathrm{set}\:\left\{\mathrm{0},\:\mathrm{1},\:\mathrm{2},\:\mathrm{3},...,\:\mathrm{10}\right\}. \\ $$$$\mathrm{Then}\:\mathrm{the}\:\mathrm{probability}\:\mathrm{that}\:\mid{x}−{y}\mid>\mathrm{5}\:\mathrm{is} \\ $$

Question Number 29176 Answers: 3 Comments: 1

∫sec x dx=?

$$\int{sec}\:{x}\:{dx}=? \\ $$

Question Number 26528 Answers: 1 Comments: 0

f length of a rectangle is reduced by 5 umits and its breadth is increasesd 3 units then area of rectangle is reduced by 8 sq units if lenghth is reduced 3 units and breadth is increased by 2 units then area of rectangle will increased by 67 sq units . then find length and breadth of rectangle

$${f}\:{length}\:\:{of}\:{a}\:{rectangle}\:{is}\:{reduced}\:{by}\:\mathrm{5}\: \\ $$$${umits}\:{and}\:{its}\:{breadth}\:{is}\:{increasesd} \\ $$$$\mathrm{3}\:{units}\:{then}\:{area}\:{of}\:{rectangle}\:{is}\:{reduced} \\ $$$${by}\:\mathrm{8}\:{sq}\:{units}\:{if}\:{lenghth}\:{is}\:{reduced} \\ $$$$\mathrm{3}\:{units}\:{and}\:{breadth}\:{is}\:{increased}\:{by}\: \\ $$$$\mathrm{2}\:{units}\:{then}\:{area}\:{of}\:{rectangle} \\ $$$${will}\:{increased}\:{by}\:\mathrm{67}\:{sq}\:{units}\:.\: \\ $$$${then}\:{find}\:{length}\:{and}\:{breadth}\:{of}\:{rectangle} \\ $$

Question Number 26523 Answers: 0 Comments: 1

The area of the regular polygon of n sides is (where R is the radius of the circumpolygon),

$$\mathrm{The}\:\mathrm{area}\:\mathrm{of}\:\mathrm{the}\:\mathrm{regular}\:\mathrm{polygon}\:\mathrm{of} \\ $$$${n}\:\mathrm{sides}\:\mathrm{is}\:\left(\mathrm{where}\:{R}\:\mathrm{is}\:\mathrm{the}\:\mathrm{radius}\:\mathrm{of}\:\mathrm{the}\right. \\ $$$$\left.\mathrm{circumpolygon}\right), \\ $$

Question Number 26513 Answers: 0 Comments: 2

let n be a fixed positive integer. How many ways are there to write n as a sum of positive integers, n=a_1 +a_2 +...+a_k with k arbitary positive integer and a_1 ≤a_2 ...≤a_k ≤a_1 +1. for example with n=4, there are four ways : 4, 2+2, 1+1+2,1+1+1+1

$$\mathrm{let}\:{n}\:\mathrm{be}\:\mathrm{a}\:\mathrm{fixed}\:\mathrm{positive}\:\mathrm{integer}.\:\mathrm{How}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{are}\:\mathrm{there}\:\mathrm{to}\:\mathrm{write}\:{n}\:\mathrm{as}\:\mathrm{a}\:\mathrm{sum}\:\mathrm{of} \\ $$$$\mathrm{positive}\:\mathrm{integers},\: \\ $$$${n}={a}_{\mathrm{1}} +{a}_{\mathrm{2}} +...+{a}_{{k}} \\ $$$$\mathrm{with}\:{k}\:\mathrm{arbitary}\:\mathrm{positive}\:\mathrm{integer}\:\mathrm{and}\:{a}_{\mathrm{1}} \leqslant{a}_{\mathrm{2}} ...\leqslant{a}_{{k}} \leqslant{a}_{\mathrm{1}} +\mathrm{1}.\:\mathrm{for}\:\mathrm{example} \\ $$$$\mathrm{with}\:{n}=\mathrm{4},\:\mathrm{there}\:\mathrm{are}\:\mathrm{four}\:\mathrm{ways}\::\:\mathrm{4},\:\mathrm{2}+\mathrm{2},\:\mathrm{1}+\mathrm{1}+\mathrm{2},\mathrm{1}+\mathrm{1}+\mathrm{1}+\mathrm{1} \\ $$

Question Number 26508 Answers: 0 Comments: 0

A={(m/n)+((8n)/m) : m, n ∈ N}, N= Natural numbers find sup(A) and inf(A)

$$\mathrm{A}=\left\{\frac{{m}}{{n}}+\frac{\mathrm{8}{n}}{{m}}\::\:{m},\:{n}\:\in\:{N}\right\},\:\mathrm{N}=\:\mathrm{Natural}\:\mathrm{numbers} \\ $$$$\mathrm{find}\:\mathrm{sup}\left(\mathrm{A}\right)\:\mathrm{and}\:\mathrm{inf}\left(\mathrm{A}\right) \\ $$

Question Number 26507 Answers: 1 Comments: 0

∫_0 ^Π ((xsinx)/(1+cos^2 x))dx

$$\overset{\Pi} {\int}_{\mathrm{0}} \frac{\mathrm{xsinx}}{\mathrm{1}+\mathrm{cos}^{\mathrm{2}} \mathrm{x}}\mathrm{dx} \\ $$

Question Number 26498 Answers: 1 Comments: 0

If (5x+7y) : (3x+11y)= 2 : 3, then x : y = ____.

$$\mathrm{If}\:\left(\mathrm{5}{x}+\mathrm{7}{y}\right)\::\:\left(\mathrm{3}{x}+\mathrm{11}{y}\right)=\:\mathrm{2}\::\:\mathrm{3},\:\mathrm{then}\: \\ $$$${x}\::\:{y}\:=\:\_\_\_\_. \\ $$

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