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Question Number 90788 Answers: 0 Comments: 1

lim_(x→−∞) (((x+2)/(x+1)))^(x/2)

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

Question Number 90787 Answers: 1 Comments: 3

Find the infinite sum Σ_(n=1) ^∞ (1/((2n−1)(2n+1)(2n+3)))

$${Find}\:{the}\:{infinite}\:{sum} \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\left(\mathrm{2}{n}−\mathrm{1}\right)\left(\mathrm{2}{n}+\mathrm{1}\right)\left(\mathrm{2}{n}+\mathrm{3}\right)} \\ $$

Question Number 90784 Answers: 0 Comments: 2

∫_0 ^2 (√((4−x)/x))−(√(x/(4−x))) dx ?

$$\underset{\mathrm{0}} {\overset{\mathrm{2}} {\int}}\:\sqrt{\frac{\mathrm{4}−{x}}{{x}}}−\sqrt{\frac{{x}}{\mathrm{4}−{x}}}\:{dx}\:? \\ $$

Question Number 90783 Answers: 0 Comments: 1

can inflection point be a max or min ?

$${can}\:{inflection}\:{point}\: \\ $$$${be}\:{a}\:{max}\:{or}\:{min}\:?\: \\ $$

Question Number 90777 Answers: 0 Comments: 0

Question Number 90773 Answers: 1 Comments: 1

Question Number 90772 Answers: 0 Comments: 2

show that the roots of the equation x^2 −2x=(b−c)^2 −1 are rational if b and c are rational numbers.

$${show}\:{that}\:{the}\:{roots}\:{of}\:{the}\:{equation} \\ $$$${x}^{\mathrm{2}} −\mathrm{2}{x}=\left({b}−{c}\right)^{\mathrm{2}} −\mathrm{1}\:{are}\:{rational}\:{if} \\ $$$${b}\:{and}\:{c}\:{are}\:{rational}\:{numbers}. \\ $$

Question Number 90770 Answers: 1 Comments: 0

(ax+by+c)dx+(px+qy+r)dy=0

$$\left({ax}+{by}+{c}\right){dx}+\left({px}+{qy}+{r}\right){dy}=\mathrm{0} \\ $$

Question Number 90762 Answers: 1 Comments: 0

2^x +2^(3x) =16 solve for x

$$\mathrm{2}^{\mathrm{x}} +\mathrm{2}^{\mathrm{3x}} =\mathrm{16} \\ $$$$\mathrm{solve}\:\mathrm{for}\:\mathrm{x} \\ $$

Question Number 90759 Answers: 0 Comments: 1

1/∫_0 ^∞ ∫_x ^∞ ((cos(t) ln^2 (x))/(t(√x)))dt dx 2/∫_(1/2) ^(3/2) ln(Γ(x))dx=((ln(π)−1)/2) 3/∫_0 ^(π/2) cos(nt) cos^m (t) dt=((πΓ(m+1))/(2^(m+1) Γ(((n+m+2)/2))Γ(((2−n+m)/2)))) 4/∫_0 ^(+∞) ((x exp(πcos(πx) sin(πsin(πx)))/(x^2 +π^2 ))dx=(π/2)exp(πexp(−π^2 ))−1)

$$\mathrm{1}/\int_{\mathrm{0}} ^{\infty} \int_{{x}} ^{\infty} \frac{{cos}\left({t}\right)\:{ln}^{\mathrm{2}} \left({x}\right)}{{t}\sqrt{{x}}}{dt}\:{dx} \\ $$$$ \\ $$$$\mathrm{2}/\int_{\mathrm{1}/\mathrm{2}} ^{\mathrm{3}/\mathrm{2}} {ln}\left(\Gamma\left({x}\right)\right){dx}=\frac{{ln}\left(\pi\right)−\mathrm{1}}{\mathrm{2}} \\ $$$$ \\ $$$$\mathrm{3}/\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {cos}\left({nt}\right)\:{cos}^{{m}} \left({t}\right)\:{dt}=\frac{\pi\Gamma\left({m}+\mathrm{1}\right)}{\mathrm{2}^{{m}+\mathrm{1}} \Gamma\left(\frac{{n}+{m}+\mathrm{2}}{\mathrm{2}}\right)\Gamma\left(\frac{\mathrm{2}−{n}+{m}}{\mathrm{2}}\right)} \\ $$$$ \\ $$$$\left.\mathrm{4}/\int_{\mathrm{0}} ^{+\infty} \frac{{x}\:{exp}\left(\pi{cos}\left(\pi{x}\right)\:{sin}\left(\pi{sin}\left(\pi{x}\right)\right)\right.}{{x}^{\mathrm{2}} +\pi^{\mathrm{2}} }{dx}=\frac{\pi}{\mathrm{2}}{exp}\left(\pi{exp}\left(−\pi^{\mathrm{2}} \right)\right)−\mathrm{1}\right) \\ $$

Question Number 90765 Answers: 0 Comments: 2

if ∫ (ln(x))^2 dx = x( ln^2 (x)+a ln(x)+b) +C a,b , C are constant. find the value of a and b

$${if}\:\int\:\left(\mathrm{ln}\left({x}\right)\right)^{\mathrm{2}} {dx}\:=\: \\ $$$${x}\left(\:\mathrm{ln}^{\mathrm{2}} \left({x}\right)+{a}\:\mathrm{ln}\left({x}\right)+{b}\right)\:+{C} \\ $$$${a},{b}\:,\:{C}\:{are}\:{constant}.\: \\ $$$${find}\:{the}\:{value}\:{of}\:{a}\:{and}\:{b}\: \\ $$

Question Number 90752 Answers: 1 Comments: 0

find ∫(1+x^2 )(√(1−x^2 ))dx

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

Question Number 90751 Answers: 0 Comments: 1

calculste lim_(n→∞) (1/n^2 )Σ_(k=1) ^n karctan((k/n))

$$\:{calculste}\:{lim}_{{n}\rightarrow\infty} \:\:\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\sum_{{k}=\mathrm{1}} ^{{n}} \:{karctan}\left(\frac{{k}}{{n}}\right) \\ $$

Question Number 90750 Answers: 0 Comments: 0

let α and β roots of x^2 −x+2=0 calculate A_n =Σ_(k=0) ^n (α^k +β^k ) B_n = Σ_(k=0) ^n (α^k −β^k )