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Question Number 39376 Answers: 0 Comments: 0

how to calculate the product (Σ_(n=0) ^∞ a_n x^n ).(Σ_(n=0) ^∞ b_n x^(2n) )?

$${how}\:{to}\:{calculate}\:{the}\:{product}\:\left(\sum_{{n}=\mathrm{0}} ^{\infty} {a}_{{n}} {x}^{{n}} \right).\left(\sum_{{n}=\mathrm{0}} ^{\infty} \:{b}_{{n}} \:{x}^{\mathrm{2}{n}} \right)? \\ $$

Question Number 39375 Answers: 0 Comments: 1

Question Number 39374 Answers: 0 Comments: 2

calculate I = ∫_0 ^∞ ((arctan(x^2 ))/(1+x^2 ))dx

$${calculate}\:{I}\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{arctan}\left({x}^{\mathrm{2}} \right)}{\mathrm{1}+{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 39373 Answers: 0 Comments: 1

find the values of integrals A = ∫_(−∞) ^(+∞) cos(x^2 +x+1)dx and B = ∫_(−∞) ^(+∞) sin(x^2 +x+1)dx

$${find}\:{the}\:{values}\:{of}\:{integrals} \\ $$$${A}\:=\:\int_{−\infty} ^{+\infty} \:{cos}\left({x}^{\mathrm{2}} \:+{x}+\mathrm{1}\right){dx}\:\:\:{and}\:{B}\:=\:\int_{−\infty} ^{+\infty} \:{sin}\left({x}^{\mathrm{2}} \:+{x}+\mathrm{1}\right){dx} \\ $$

Question Number 39371 Answers: 2 Comments: 1

Question Number 39370 Answers: 0 Comments: 1

let I (λ) = ∫_(−∞) ^(+∞) ((cos(λx))/((1+ix)^2 ))dx 1) extract Re(I(λ)) and Im(I(λ)) 2) calculate I(λ) 3) conclude the values of Re(I(λ)) and Im(I(λ)).

$${let}\:{I}\:\left(\lambda\right)\:=\:\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{cos}\left(\lambda{x}\right)}{\left(\mathrm{1}+{ix}\right)^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{1}\right)\:\:{extract}\:{Re}\left({I}\left(\lambda\right)\right)\:{and}\:{Im}\left({I}\left(\lambda\right)\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{I}\left(\lambda\right) \\ $$$$\left.\mathrm{3}\right)\:{conclude}\:\:{the}\:{values}\:{of}\:{Re}\left({I}\left(\lambda\right)\right)\:{and}\:{Im}\left({I}\left(\lambda\right)\right). \\ $$

Question Number 39369 Answers: 0 Comments: 1

1) calculate F(x)= ∫_1 ^(√x) ((arctan(t))/t^2 )dt with x≥1 2) calculate A_n = ∫_1 ^(√n) ((arctan(t))/t^2 ) dt and find lim_(n→+∞) A_n

$$\left.\mathrm{1}\right)\:{calculate}\:{F}\left({x}\right)=\:\int_{\mathrm{1}} ^{\sqrt{{x}}} \:\:\:\frac{{arctan}\left({t}\right)}{{t}^{\mathrm{2}} }{dt}\:\:\:{with}\:{x}\geqslant\mathrm{1} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\:{A}_{{n}} =\:\int_{\mathrm{1}} ^{\sqrt{{n}}} \:\:\frac{{arctan}\left({t}\right)}{{t}^{\mathrm{2}} }\:{dt}\:\:{and}\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} \\ $$

Question Number 39368 Answers: 0 Comments: 1

let F(t)= ∫_0 ^(+∞) ((sinx)/(x(1+x^2 ))) e^(−tx(1+x^2 )) dx witht≥0 1) caculate (dF/dt)(t) 2) find a simple form of F(t) 3) find the value of ∫_0 ^∞ ((sinx)/(x(1+x^2 )dx )).

$${let}\:{F}\left({t}\right)=\:\int_{\mathrm{0}} ^{+\infty} \:\:\:\frac{{sinx}}{{x}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}\:{e}^{−{tx}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)} {dx}\:\:{witht}\geqslant\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{caculate}\:\:\frac{{dF}}{{dt}}\left({t}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{a}\:{simple}\:{form}\:{of}\:{F}\left({t}\right) \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{sinx}}{{x}\left(\mathrm{1}+{x}^{\mathrm{2}} \right){dx}\:}. \\ $$

Question Number 39365 Answers: 0 Comments: 1

Question Number 39357 Answers: 0 Comments: 0

∫ (1/(xln(x+1))) dx

$$\int\:\frac{\mathrm{1}}{{xln}\left({x}+\mathrm{1}\right)}\:{dx} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 39349 Answers: 1 Comments: 1

Question Number 39346 Answers: 0 Comments: 2

Question Number 39342 Answers: 1 Comments: 0

Let f(x) = ∫_(0 ) ^2 ∣x−t∣ dt (x>0) , then minimum value of f(x) is ?

$$\mathrm{Let}\:\mathrm{f}\left({x}\right)\:=\:\int_{\mathrm{0}\:} ^{\mathrm{2}} \:\mid{x}−{t}\mid\:\mathrm{dt}\:\left({x}>\mathrm{0}\right)\:,\:\mathrm{then} \\ $$$$\mathrm{minimum}\:\mathrm{value}\:\mathrm{of}\:\mathrm{f}\left({x}\right)\:\mathrm{is}\:? \\ $$

Question Number 39341 Answers: 0 Comments: 4

I_1 = ∫_0 ^π ((sin 884x sin 1122x)/(sin x)) dx I_2 = ∫_0 ^1 ((x^(238) (x^(1768) −1))/((x^2 −1))) dx then value of (I_1 /I_2 ) =?

$$\mathrm{I}_{\mathrm{1}} =\:\int_{\mathrm{0}} ^{\pi} \frac{\mathrm{sin}\:\mathrm{884}{x}\:\mathrm{sin}\:\mathrm{1122}{x}}{\mathrm{sin}\:{x}}\:{dx} \\ $$$$\mathrm{I}_{\mathrm{2}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{x}^{\mathrm{238}} \left({x}^{\mathrm{1768}} −\mathrm{1}\right)}{\left({x}^{\mathrm{2}} −\mathrm{1}\right)}\:{dx} \\ $$$${then}\:{value}\:{of}\:\frac{\mathrm{I}_{\mathrm{1}} }{\mathrm{I}_{\mathrm{2}} }\:=? \\ $$

Question Number 39338 Answers: 2 Comments: 2

If f(x) = ∫_0 ^4 e^(∣t−x∣) dt (0≤x≤4), maximum value of f(x) is = ?

$$\mathrm{If}\:\mathrm{f}\left({x}\right)\:=\:\int_{\mathrm{0}} ^{\mathrm{4}} \:\mathrm{e}^{\mid\mathrm{t}−{x}\mid} \:\mathrm{dt}\:\:\:\:\left(\mathrm{0}\leqslant{x}\leqslant\mathrm{4}\right), \\ $$$$\mathrm{maximum}\:\mathrm{value}\:\mathrm{of}\:\mathrm{f}\left({x}\right)\:\mathrm{is}\:=\:? \\ $$

Question Number 39336 Answers: 0 Comments: 5

Question Number 39337 Answers: 0 Comments: 0

tan^(−1) 2 + tan^(−1) 3=cosec^(−1) x ,then x is equal to (a) 4 (b) (√2) (d) −(√2) (d) none of these

$${tan}^{−\mathrm{1}} \:\mathrm{2}\:+\:{tan}^{−\mathrm{1}} \:\mathrm{3}={cosec}^{−\mathrm{1}} {x}\:\:,{then} \\ $$$${x}\:\:{is}\:\:{equal}\:\:{to} \\ $$$$\:\left({a}\right)\:\:\:\mathrm{4}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left({b}\right)\:\:\sqrt{\mathrm{2}}\:\: \\ $$$$\left({d}\right)\:−\sqrt{\mathrm{2}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left({d}\right)\:{none}\:{of}\:{these} \\ $$

Question Number 39332 Answers: 0 Comments: 1

1) simplify S_n (x)=Σ_(k=1) ^n sin^2 (kx) 2)simplify A_n =Σ_(k=1) ^n sin^2 (((kπ)/n))

$$\left.\mathrm{1}\right)\:{simplify}\:{S}_{{n}} \left({x}\right)=\sum_{{k}=\mathrm{1}} ^{{n}} \:{sin}^{\mathrm{2}} \left({kx}\right) \\ $$$$\left.\mathrm{2}\right){simplify}\:\:{A}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:{sin}^{\mathrm{2}} \left(\frac{{k}\pi}{{n}}\right) \\ $$

Question Number 39404 Answers: 1 Comments: 7

Question Number 39312 Answers: 5 Comments: 0

prove that (tan 4a+tan 2a)(1−tan^2 3a tan^2 a)=2tan 3a sec^2 a

$${prove}\:{that} \\ $$$$\left({tan}\:\mathrm{4}{a}+{tan}\:\mathrm{2}{a}\right)\left(\mathrm{1}−{tan}^{\mathrm{2}} \mathrm{3}{a}\:{tan}^{\mathrm{2}} {a}\right)=\mathrm{2}{tan}\:\mathrm{3}{a}\:{sec}^{\mathrm{2}} {a} \\ $$

Question Number 39303 Answers: 1 Comments: 1

Question Number 39300 Answers: 1 Comments: 0

Question Number 39298 Answers: 0 Comments: 0

Question Number 39292 Answers: 0 Comments: 3

let f(x)= arctan(2x) 1) calculate f^((n)) (x) then f^((n)) (0) 2) developp f at integr serie 3) let F(t)= ∫_0 ^t arctan(2x)dx developp F at integr serie 4) give F(1) at form of serie.

$${let}\:{f}\left({x}\right)=\:{arctan}\left(\mathrm{2}{x}\right) \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{\left({n}\right)} \left({x}\right)\:{then}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right)\:{developp}\:{f}\:{at}\:{integr}\:{serie} \\ $$$$\left.\mathrm{3}\right)\:{let}\:{F}\left({t}\right)=\:\int_{\mathrm{0}} ^{{t}} \:\:{arctan}\left(\mathrm{2}{x}\right){dx} \\ $$$${developp}\:{F}\:{at}\:{integr}\:{serie} \\ $$$$\left.\mathrm{4}\right)\:{give}\:{F}\left(\mathrm{1}\right)\:{at}\:{form}\:{of}\:{serie}. \\ $$

Question Number 39289 Answers: 1 Comments: 0

Find the value of x if the matrix (((3x 5x)),((x 3)) ) (((x 1)),((3 x)) ) has no inverse

$${Find}\:{the}\:{value}\:{of}\:{x}\:{if}\: \\ $$$${the}\:{matrix}\: \\ $$$$ \\ $$$$\begin{pmatrix}{\mathrm{3}{x}\:\:\:\:\:\:\:\:\mathrm{5}{x}}\\{{x}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{3}}\end{pmatrix}\begin{pmatrix}{{x}\:\:\:\:\:\:\mathrm{1}}\\{\mathrm{3}\:\:\:\:\:\:\:\:\:{x}}\end{pmatrix}\: \\ $$$${has}\:{no}\:{inverse} \\ $$

Question Number 39291 Answers: 0 Comments: 2

let f(x)=(e^(−x^2 ) /(1+x^2 )) developp f at integr serie .

$${let}\:{f}\left({x}\right)=\frac{{e}^{−{x}^{\mathrm{2}} } }{\mathrm{1}+{x}^{\mathrm{2}} } \\ $$$${developp}\:{f}\:{at}\:{integr}\:{serie}\:. \\ $$

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