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Question Number 105257 Answers: 2 Comments: 0

y′′−2y′+y = xe^x sin x

$${y}''−\mathrm{2}{y}'+{y}\:=\:{xe}^{{x}} \mathrm{sin}\:{x}\: \\ $$

Question Number 105256 Answers: 0 Comments: 0

Question Number 101848 Answers: 2 Comments: 0

(cos x) (dy/dx) +y sin x = 2x cos^2 x , y((π/4)) = ((−15π^2 (√2))/(32))

$$\left(\mathrm{cos}\:\mathrm{x}\right)\:\frac{\mathrm{dy}}{\mathrm{dx}}\:+\mathrm{y}\:\mathrm{sin}\:\mathrm{x}\:=\:\mathrm{2x}\:\mathrm{cos}\:^{\mathrm{2}} \mathrm{x}\:, \\ $$$$\mathrm{y}\left(\frac{\pi}{\mathrm{4}}\right)\:=\:\frac{−\mathrm{15}\pi^{\mathrm{2}} \sqrt{\mathrm{2}}}{\mathrm{32}} \\ $$

Question Number 101846 Answers: 2 Comments: 0

Question Number 101841 Answers: 1 Comments: 0

xy′ + y = y^2

$${xy}'\:+\:{y}\:=\:{y}^{\mathrm{2}} \\ $$

Question Number 101835 Answers: 3 Comments: 0

∫_0 ^∞ (1/(e^x +1)) dx

$$\underset{\mathrm{0}} {\overset{\infty} {\int}}\frac{\mathrm{1}}{{e}^{{x}} +\mathrm{1}}\:{dx}\: \\ $$

Question Number 101833 Answers: 3 Comments: 0

∫ _(−1)^1 (√((1+x)/(1−x))) dx ?

$$\int\:_{−\mathrm{1}} ^{\mathrm{1}} \sqrt{\frac{\mathrm{1}+{x}}{\mathrm{1}−{x}}}\:{dx}\:?\: \\ $$

Question Number 101832 Answers: 0 Comments: 1

∫((ln x)/(x^2 +1)) dx ? (JS ⊛)

$$\int\frac{\mathrm{ln}\:{x}}{{x}^{\mathrm{2}} +\mathrm{1}}\:{dx}\:?\: \\ $$$$\left({JS}\:\circledast\right) \\ $$

Question Number 101828 Answers: 2 Comments: 0

∫_0 ^∞ ((e^(πx) −e^x )/(x(e^(πx) +1)(e^x +1)))dx

$$\underset{\mathrm{0}} {\overset{\infty} {\int}}\frac{\mathrm{e}^{\pi\mathrm{x}} −\mathrm{e}^{\mathrm{x}} }{\mathrm{x}\left(\mathrm{e}^{\pi\mathrm{x}} +\mathrm{1}\right)\left(\mathrm{e}^{\mathrm{x}} +\mathrm{1}\right)}\mathrm{dx} \\ $$

Question Number 101822 Answers: 1 Comments: 0

lim_(n→∞) ((φ^(n+1) −(−φ)^(−n−1) )/(φ^n −(−φ)^(−n) )) = (JS ⊛)

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{\phi^{{n}+\mathrm{1}} −\left(−\phi\right)^{−{n}−\mathrm{1}} }{\phi^{{n}} −\left(−\phi\right)^{−{n}} }\:=\: \\ $$$$\left({JS}\:\circledast\right) \\ $$

Question Number 101816 Answers: 1 Comments: 0

∫_1 ^( e) (((tan^(−1) x)/x)+((log)/(x^2 +1)))dx

$$\int_{\mathrm{1}} ^{\:\:{e}} \left(\frac{{tan}^{−\mathrm{1}} {x}}{{x}}+\frac{{log}}{{x}^{\mathrm{2}} +\mathrm{1}}\right){dx} \\ $$

Question Number 101808 Answers: 2 Comments: 0

∫_(1/3) ^1 (((x−x^3 )^(1/3) )/x^4 )dx

$$\int_{\frac{\mathrm{1}}{\mathrm{3}}} ^{\mathrm{1}} \frac{\left(\mathrm{x}−\mathrm{x}^{\mathrm{3}} \right)^{\mathrm{1}/\mathrm{3}} }{\mathrm{x}^{\mathrm{4}} }\mathrm{dx} \\ $$

Question Number 101803 Answers: 1 Comments: 2

((√2)−1)^x +((√2)+1)^x =((√6))^x

$$\left(\sqrt{\mathrm{2}}−\mathrm{1}\right)^{{x}} +\left(\sqrt{\mathrm{2}}+\mathrm{1}\right)^{{x}} =\left(\sqrt{\mathrm{6}}\right)^{{x}} \\ $$

Question Number 101800 Answers: 1 Comments: 3

Question Number 101794 Answers: 1 Comments: 0

Question Number 101793 Answers: 1 Comments: 0

∫_1 ^2 ((logu)/(((√(u−1)))((√(u−1))+1)))du

$$\int_{\mathrm{1}} ^{\mathrm{2}} \frac{{logu}}{\left(\sqrt{{u}−\mathrm{1}}\right)\left(\sqrt{{u}−\mathrm{1}}+\mathrm{1}\right)}{du} \\ $$

Question Number 101791 Answers: 2 Comments: 0

(1/n^(3 ) )lim_(n→∞) (ne^(−((1/n))^2 ) +2ne^(−((2/n))^2 ) +....∞)

$$\frac{\mathrm{1}}{{n}^{\mathrm{3}\:\:} }\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left({ne}^{−\left(\frac{\mathrm{1}}{{n}}\right)^{\mathrm{2}} } +\mathrm{2}{ne}^{−\left(\frac{\mathrm{2}}{{n}}\right)^{\mathrm{2}} } +....\infty\right) \\ $$

Question Number 101783 Answers: 2 Comments: 0

∫_( 0) ^( 1) ((ln(x^2 + 1))/(x + 1))

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

Question Number 101779 Answers: 1 Comments: 0

If S_n is the sum of the first n terms of an A.P. Express S_(2k) in terms of S_k and S_(3k)

$$\mathrm{If}\:\:\:\:\mathrm{S}_{\mathrm{n}} \:\:\mathrm{is}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{first}\:\:\mathrm{n}\:\:\mathrm{terms}\:\mathrm{of}\:\mathrm{an}\:\mathrm{A}.\mathrm{P}. \\ $$$$\mathrm{Express}\:\:\:\mathrm{S}_{\mathrm{2k}} \:\:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:\:\:\mathrm{S}_{\mathrm{k}} \:\:\mathrm{and}\:\:\:\mathrm{S}_{\mathrm{3k}} \\ $$

Question Number 101778 Answers: 2 Comments: 0

Question Number 101777 Answers: 0 Comments: 0

Find a curve passing through pointA(0;1) for which the triangle formed by the axis Oy ,tangent to the curve at its arbitrary point and the radius−vector of the point of contact,issosceles(and base is the segment of the tangent from the point of contact to the axis Oy)

Question Number 101775 Answers: 0 Comments: 1

∫x^x^x ∙x^x ∙xdx=? or it able to solve?

$$\int{x}^{{x}^{{x}} } \centerdot{x}^{{x}} \centerdot{xdx}=? \\ $$$${or}\:{it}\:{able}\:{to}\:{solve}? \\ $$

Question Number 101770 Answers: 1 Comments: 2

lim_(n→∞) ((1^(13) +2^(13) +3^(13) +4^(13) +...+n^(13) )/n^(14) ) ?

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{1}^{\mathrm{13}} +\mathrm{2}^{\mathrm{13}} +\mathrm{3}^{\mathrm{13}} +\mathrm{4}^{\mathrm{13}} +...+{n}^{\mathrm{13}} }{{n}^{\mathrm{14}} }\:? \\ $$

Question Number 101768 Answers: 0 Comments: 1

Question Number 101767 Answers: 0 Comments: 2

Question Number 101766 Answers: 0 Comments: 1

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