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Question Number 65690 Answers: 0 Comments: 2

calculate ∫_0 ^1 ((ln^2 (x))/(1+x^2 ))dx

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

Question Number 65687 Answers: 0 Comments: 0

∫_0 ^(2π) ((sin(3t))/(5−3cos(t))) dt=0 using Residue theorem

$$\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\frac{{sin}\left(\mathrm{3}{t}\right)}{\mathrm{5}−\mathrm{3}{cos}\left({t}\right)}\:{dt}=\mathrm{0}\:\mathrm{using}\:\:\mathrm{Residue}\:\mathrm{theorem} \\ $$

Question Number 65683 Answers: 0 Comments: 0

Question Number 65681 Answers: 1 Comments: 0

∫_0 ^1 (Π_(r=1) ^n (x+r))(Σ_(k=1) ^n (1/(x+k))) dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \left(\underset{{r}=\mathrm{1}} {\overset{{n}} {\prod}}\left({x}+{r}\right)\right)\left(\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\:\frac{\mathrm{1}}{{x}+{k}}\right)\:{dx} \\ $$

Question Number 65680 Answers: 1 Comments: 0

Question Number 65679 Answers: 0 Comments: 2

let A_n =∫_(−∞) ^(+∞) ((cos(2^n x))/((x^2 +3)^2 ))dx 1) calculate A_n interms of n 2)find nsture of the serie ΣA_n and Σn^n A_n

$${let}\:\:{A}_{{n}} =\int_{−\infty} ^{+\infty} \:\:\:\frac{{cos}\left(\mathrm{2}^{{n}} {x}\right)}{\left({x}^{\mathrm{2}} +\mathrm{3}\right)^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{A}_{{n}} \:{interms}\:{of}\:{n} \\ $$$$\left.\mathrm{2}\right){find}\:{nsture}\:{of}\:{the}\:{serie}\:\Sigma{A}_{{n}} \:\:\:\:{and}\:\Sigma{n}^{{n}} \:{A}_{{n}} \\ $$

Question Number 65678 Answers: 1 Comments: 1

calculate ∫ ((3x+1)/((x^2 −4)(x^3 +2x−3)))dx

$${calculate}\:\:\int\:\:\:\:\frac{\mathrm{3}{x}+\mathrm{1}}{\left({x}^{\mathrm{2}} −\mathrm{4}\right)\left({x}^{\mathrm{3}} +\mathrm{2}{x}−\mathrm{3}\right)}{dx} \\ $$

Question Number 65677 Answers: 0 Comments: 0

let S_n =Σ_(k=1) ^n (1/(√(k^2 +k))) 1) find a equivalent of S_n when n→+∞ 2)prove that (S_n ) is convergent.

$${let}\:{S}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{\sqrt{{k}^{\mathrm{2}} +{k}}} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{equivalent}\:{of}\:{S}_{{n}} \:{when}\:{n}\rightarrow+\infty \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:\left({S}_{{n}} \right)\:{is}\:{convergent}. \\ $$

Question Number 65676 Answers: 0 Comments: 1

calculate ∫_0 ^∞ e^(−x^2 −(1/x^2 )) dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{x}^{\mathrm{2}} −\frac{\mathrm{1}}{{x}^{\mathrm{2}} }} {dx} \\ $$

Question Number 65675 Answers: 0 Comments: 0

1) find ∫ (dx/(√((x^2 +1)(x−2))))

$$\left.\mathrm{1}\right)\:{find}\:\int\:\frac{{dx}}{\sqrt{\left({x}^{\mathrm{2}} +\mathrm{1}\right)\left({x}−\mathrm{2}\right)}} \\ $$

Question Number 65674 Answers: 0 Comments: 1

find the value of Σ_(n=2) ^∞ (n/((n+1)^2 (n−1)^3 ))

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

Question Number 65673 Answers: 0 Comments: 1

find Σ_(n=2) ^∞ (((−1)^n )/((n^2 −1)^2 ))

$${find}\:\sum_{{n}=\mathrm{2}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{n}} }{\left({n}^{\mathrm{2}} −\mathrm{1}\right)^{\mathrm{2}} } \\ $$

Question Number 65665 Answers: 0 Comments: 1

calculate ∫_(−2) ^(+∞) (e^(−x) /(√(x+2))) dx

$${calculate}\:\int_{−\mathrm{2}} ^{+\infty} \:\:\frac{{e}^{−{x}} }{\sqrt{{x}+\mathrm{2}}}\:{dx} \\ $$$$ \\ $$

Question Number 65664 Answers: 1 Comments: 0

solve (((√(1−x))−(√(2x+1)))/((√(1−x))+(√(2x+1)))) =((x+1)/3)

$${solve}\:\frac{\sqrt{\mathrm{1}−{x}}−\sqrt{\mathrm{2}{x}+\mathrm{1}}}{\sqrt{\mathrm{1}−{x}}+\sqrt{\mathrm{2}{x}+\mathrm{1}}}\:=\frac{{x}+\mathrm{1}}{\mathrm{3}} \\ $$

Question Number 65724 Answers: 2 Comments: 0

Question Number 65723 Answers: 1 Comments: 0

Question Number 65700 Answers: 0 Comments: 1

Question Number 65651 Answers: 0 Comments: 1

Question Number 65704 Answers: 0 Comments: 1

∫ln^2 xsin(x)dx

$$\int{ln}^{\mathrm{2}} {xsin}\left({x}\right){dx} \\ $$

Question Number 65601 Answers: 0 Comments: 1

1.If y=x^(n−1) log x,then prove that,x^2 (d^2 y/dx^2 )+(3−2n)x(dy/dx)+(n−1)^2 y=0 2.If ((mtan (α−θ))/(cos^2 θ))=((ntan θ)/(cos^2 (α−θ))),then prove that,θ=(1/2)[α−tan^(−1) (((n−m)/(n+m))tan α)]

$$\mathrm{1}.\mathrm{If}\:\boldsymbol{{y}}=\boldsymbol{{x}}^{\boldsymbol{{n}}−\mathrm{1}} \mathrm{log}\:\boldsymbol{{x}},\mathrm{then}\:\mathrm{prove}\:\mathrm{that},\boldsymbol{{x}}^{\mathrm{2}} \frac{\boldsymbol{{d}}^{\mathrm{2}} \boldsymbol{{y}}}{\boldsymbol{{dx}}^{\mathrm{2}} }+\left(\mathrm{3}−\mathrm{2}\boldsymbol{{n}}\right)\boldsymbol{{x}}\frac{\boldsymbol{{dy}}}{\boldsymbol{{dx}}}+\left(\boldsymbol{{n}}−\mathrm{1}\right)^{\mathrm{2}} \boldsymbol{{y}}=\mathrm{0} \\ $$$$\mathrm{2}.\boldsymbol{\mathrm{I}}\mathrm{f}\:\frac{\mathrm{mtan}\:\left(\alpha−\theta\right)}{\mathrm{cos}\:^{\mathrm{2}} \theta}=\frac{{n}\mathrm{tan}\:\theta}{\mathrm{cos}\:^{\mathrm{2}} \left(\alpha−\theta\right)},\mathrm{then}\:\mathrm{prove}\:\mathrm{that},\theta=\frac{\mathrm{1}}{\mathrm{2}}\left[\alpha−\mathrm{tan}^{−\mathrm{1}} \left(\frac{{n}−{m}}{{n}+{m}}\mathrm{tan}\:\alpha\right)\right] \\ $$

Question Number 65593 Answers: 1 Comments: 5

∫_0 ^(1/3) (3x + 1)^5 dx =

$$\int_{\mathrm{0}} ^{\frac{\mathrm{1}}{\mathrm{3}}} \left(\mathrm{3}{x}\:+\:\mathrm{1}\right)^{\mathrm{5}} {dx}\:= \\ $$

Question Number 65592 Answers: 1 Comments: 0

∫_0 ^(1/3) (3x + 1)^5 dx =

$$\int_{\mathrm{0}} ^{\frac{\mathrm{1}}{\mathrm{3}}} \left(\mathrm{3}{x}\:+\:\mathrm{1}\right)^{\mathrm{5}} {dx}\:= \\ $$

Question Number 65587 Answers: 1 Comments: 4

Prove that 1^3 + 2^3 + 3^3 + … + n^3 = (1+2+3+...+n)^2

$${Prove}\:\:{that} \\ $$$$\:\:\:\:\:\mathrm{1}^{\mathrm{3}} \:+\:\mathrm{2}^{\mathrm{3}} \:+\:\mathrm{3}^{\mathrm{3}} \:+\:\ldots\:+\:{n}^{\mathrm{3}} \:\:=\:\:\left(\mathrm{1}+\mathrm{2}+\mathrm{3}+...+{n}\right)^{\mathrm{2}} \\ $$

Question Number 65589 Answers: 0 Comments: 8

Question Number 65581 Answers: 0 Comments: 1

Question Number 65580 Answers: 0 Comments: 0

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