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Question Number 57820 Answers: 0 Comments: 1

solve (√(x+1))y^′ −(√(x−2))y =x^2 e^(−2x) with y(3) =1

$${solve}\:\sqrt{{x}+\mathrm{1}}{y}^{'} −\sqrt{{x}−\mathrm{2}}{y}\:={x}^{\mathrm{2}} \:{e}^{−\mathrm{2}{x}} \:\:\:{with}\:{y}\left(\mathrm{3}\right)\:=\mathrm{1} \\ $$

Question Number 57819 Answers: 2 Comments: 7

a. ∫ [((1−e^x )/(1+e^x ))]^(1/2) dx=? b. ∫ ((lnx)/(√(1+x)))=? c. ∫_( (√e)) ^( e) sin(lnx)dx=?

Question Number 57818 Answers: 1 Comments: 0

Question Number 57817 Answers: 1 Comments: 1

find the value of ∫_(π/3) ^(π/2) (dx/(√(2cos^2 x +3sin^2 x)))

$$\:{find}\:{the}\:{value}\:{of}\:\:\int_{\frac{\pi}{\mathrm{3}}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{dx}}{\sqrt{\mathrm{2}{cos}^{\mathrm{2}} {x}\:+\mathrm{3}{sin}^{\mathrm{2}} {x}}} \\ $$

Question Number 57805 Answers: 1 Comments: 2

Find the image of y=3x+1 under the mapping (((2 3)),((1 2)) ).

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{image}\:\mathrm{of}\:\mathrm{y}=\mathrm{3x}+\mathrm{1}\:\mathrm{under}\:\mathrm{the} \\ $$$$\mathrm{mapping}\:\begin{pmatrix}{\mathrm{2}\:\:\:\mathrm{3}}\\{\mathrm{1}\:\:\:\mathrm{2}}\end{pmatrix}. \\ $$

Question Number 57789 Answers: 0 Comments: 0

Question Number 57791 Answers: 3 Comments: 0

If a + b + c = 1 a^2 + b^2 + c^2 = 2 a^3 + b^3 + c^3 = 3 then a^5 + b^5 + c^(5 ) = ?

$$\:\mathrm{If}\:\:\:\:\:\mathrm{a}\:+\:\mathrm{b}\:+\:\mathrm{c}\:\:=\:\:\mathrm{1}\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{a}^{\mathrm{2}} \:+\:\mathrm{b}^{\mathrm{2}} \:+\:\mathrm{c}^{\mathrm{2}} \:\:=\:\:\mathrm{2} \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{a}^{\mathrm{3}} \:+\:\mathrm{b}^{\mathrm{3}} \:+\:\mathrm{c}^{\mathrm{3}} \:\:=\:\:\mathrm{3}\:\: \\ $$$$\mathrm{then}\:\:\:\:\:\:\mathrm{a}^{\mathrm{5}} \:+\:\mathrm{b}^{\mathrm{5}} \:+\:\mathrm{c}^{\mathrm{5}\:\:} =\:\:? \\ $$

Question Number 57790 Answers: 1 Comments: 0

Given N= [((5 3)),((6 4)) ]and P= [((4 −3)),((−6 5)) ], find NP and deduce the inverse of P.

$$\mathrm{Given}\:\mathrm{N}=\begin{bmatrix}{\mathrm{5}\:\:\:\:\:\:\mathrm{3}}\\{\mathrm{6}\:\:\:\:\:\:\:\mathrm{4}}\end{bmatrix}\mathrm{and}\:\mathrm{P}=\begin{bmatrix}{\mathrm{4}\:\:\:\:\:−\mathrm{3}}\\{−\mathrm{6}\:\:\:\:\mathrm{5}}\end{bmatrix}, \\ $$$$\mathrm{find}\:\mathrm{NP}\:\mathrm{and}\:\mathrm{deduce}\:\mathrm{the}\:\mathrm{inverse}\:\mathrm{of}\:\mathrm{P}. \\ $$

Question Number 57785 Answers: 1 Comments: 2

Question Number 57784 Answers: 0 Comments: 0

Question Number 57783 Answers: 1 Comments: 0

Question Number 57779 Answers: 0 Comments: 0

kno_3 ⇒k_2 o+n_2 +o_2

$${kno}_{\mathrm{3}} \Rightarrow{k}_{\mathrm{2}} {o}+{n}_{\mathrm{2}} +{o}_{\mathrm{2}} \\ $$

Question Number 57770 Answers: 1 Comments: 0

Question Number 57754 Answers: 2 Comments: 1

f(x)=ln(x) (f○f)′=?

$$\boldsymbol{{f}}\left(\boldsymbol{{x}}\right)=\boldsymbol{{ln}}\left(\boldsymbol{{x}}\right) \\ $$$$\left(\boldsymbol{{f}}\circ\boldsymbol{{f}}\right)'=? \\ $$

Question Number 57750 Answers: 1 Comments: 0

find ∫ x^2 (√(25−x^2 ))dx

$${find}\:\int\:{x}^{\mathrm{2}} \sqrt{\mathrm{25}−{x}^{\mathrm{2}} }{dx}\: \\ $$

Question Number 57749 Answers: 1 Comments: 3

find ∫ (dx/(x^2 (√(9+x^2 ))))

$${find}\:\int\:\:\frac{{dx}}{{x}^{\mathrm{2}} \sqrt{\mathrm{9}+{x}^{\mathrm{2}} }} \\ $$

Question Number 57748 Answers: 2 Comments: 2

find ∫ x^2 (√(4+x^2 ))dx

$${find}\:\int\:{x}^{\mathrm{2}} \sqrt{\mathrm{4}+{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 57746 Answers: 0 Comments: 4

let f(x)=∫_(−∞) ^(+∞) (dt/((t^2 −2xt +1)^2 )) with ∣x∣<1 (x real) 1) determine a explicit form for f(x) 2) find also g(x) =∫_(−∞) ^(+∞) ((tdt)/((t^2 −2xt +1)^3 )) 3) calculate ∫_(−∞) ^(+∞) (dt/((t^2 −(√2)t +1)^2 )) and ∫_(−∞) ^(+∞) ((tdt)/((t^2 −(√2)t +1)^3 )) 4) calculate A(θ) =∫_(−∞) ^(+∞) (dt/((t^2 −2cosθ t+1)^2 )) and B(θ) =∫_(−∞) ^(+∞) ((tdt)/((t^2 −2cosθ t +1)^3 )) with 0<θ <(π/2) .

$${let}\:{f}\left({x}\right)=\int_{−\infty} ^{+\infty} \:\:\:\:\frac{{dt}}{\left({t}^{\mathrm{2}} −\mathrm{2}{xt}\:+\mathrm{1}\right)^{\mathrm{2}} }\:\:{with}\:\mid{x}\mid<\mathrm{1}\:\:\:\left({x}\:{real}\right) \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{a}\:{explicit}\:{form}\:\:{for}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{also}\:{g}\left({x}\right)\:=\int_{−\infty} ^{+\infty} \:\:\:\frac{{tdt}}{\left({t}^{\mathrm{2}} −\mathrm{2}{xt}\:+\mathrm{1}\right)^{\mathrm{3}} } \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{dt}}{\left({t}^{\mathrm{2}} −\sqrt{\mathrm{2}}{t}\:+\mathrm{1}\right)^{\mathrm{2}} }\:\:\:{and}\:\int_{−\infty} ^{+\infty} \:\:\frac{{tdt}}{\left({t}^{\mathrm{2}} −\sqrt{\mathrm{2}}{t}\:+\mathrm{1}\right)^{\mathrm{3}} } \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:{A}\left(\theta\right)\:=\int_{−\infty} ^{+\infty} \:\:\:\frac{{dt}}{\left({t}^{\mathrm{2}} \:−\mathrm{2}{cos}\theta\:{t}+\mathrm{1}\right)^{\mathrm{2}} }\:\:\:{and}\: \\ $$$${B}\left(\theta\right)\:=\int_{−\infty} ^{+\infty} \:\:\:\:\frac{{tdt}}{\left({t}^{\mathrm{2}} \:−\mathrm{2}{cos}\theta\:{t}\:+\mathrm{1}\right)^{\mathrm{3}} }\:\:\:\:{with}\:\mathrm{0}<\theta\:<\frac{\pi}{\mathrm{2}}\:\:\:\:\:. \\ $$

Question Number 57736 Answers: 1 Comments: 0

Can we use L′Ho^ pital′s rule if we have a fraction in the form (+∞)/(−∞) ?

$${Can}\:{we}\:{use}\:{L}'{H}\hat {{o}pital}'{s}\:{rule}\:{if}\:{we}\:{have} \\ $$$${a}\:{fraction}\:{in}\:{the}\:{form}\:\left(+\infty\right)/\left(−\infty\right)\:\:\:? \\ $$

Question Number 57735 Answers: 2 Comments: 0