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Question Number 31133 Answers: 0 Comments: 3

Question Number 31125 Answers: 2 Comments: 1

Let n be a positive integer. Then x^2 + 1 is a factor of (x^4 + 3)^n − [(x^2 + 3)(x^2 − 1)]^n for ... (A) All n (B) Odd n (C) Even n (D) n ≥ 3 (E) None of these options

$$\mathrm{Let}\:{n}\:\mathrm{be}\:\mathrm{a}\:\mathrm{positive}\:\mathrm{integer}.\:\mathrm{Then}\:{x}^{\mathrm{2}} \:+\:\mathrm{1}\: \\ $$$$\mathrm{is}\:\mathrm{a}\:\mathrm{factor}\:\mathrm{of}\:\left({x}^{\mathrm{4}} \:+\:\mathrm{3}\right)^{{n}} \:−\:\left[\left({x}^{\mathrm{2}} \:+\:\mathrm{3}\right)\left({x}^{\mathrm{2}} \:−\:\mathrm{1}\right)\right]^{{n}} \\ $$$$\mathrm{for}\:... \\ $$$$\left(\mathrm{A}\right)\:\mathrm{All}\:{n} \\ $$$$\left(\mathrm{B}\right)\:\mathrm{Odd}\:{n} \\ $$$$\left(\mathrm{C}\right)\:\mathrm{Even}\:{n} \\ $$$$\left(\mathrm{D}\right)\:{n}\:\geqslant\:\mathrm{3} \\ $$$$\left(\mathrm{E}\right)\:\mathrm{None}\:\mathrm{of}\:\mathrm{these}\:\mathrm{options} \\ $$

Question Number 31118 Answers: 1 Comments: 1

Question Number 31114 Answers: 1 Comments: 2

Question Number 31113 Answers: 2 Comments: 0

Two lines through the point (1,−3) are tamgent to the curve y=x^2 . Find the equation of these two lines and make a sketch to verify your results.

$${Two}\:{lines}\:{through}\:{the}\:{point}\:\left(\mathrm{1},−\mathrm{3}\right) \\ $$$${are}\:{tamgent}\:{to}\:{the}\:{curve}\:{y}={x}^{\mathrm{2}} . \\ $$$${Find}\:{the}\:{equation}\:{of}\:{these}\:{two} \\ $$$${lines}\:{and}\:{make}\:{a}\:{sketch}\:{to}\:{verify} \\ $$$${your}\:{results}. \\ $$

Question Number 31111 Answers: 1 Comments: 1

Question Number 31109 Answers: 0 Comments: 5

a^4 + b^4 + 13 is a possible largest prime number . a and b are prime numbers . Find a and b .

$$\boldsymbol{{a}}^{\mathrm{4}} \:+\:\boldsymbol{{b}}^{\mathrm{4}} \:+\:\mathrm{13}\:\:\:{is}\:\:{a}\:\:{possible}\:\:{largest}\:\:{prime}\:\:{number}\:. \\ $$$$\boldsymbol{{a}}\:\:{and}\:\:\boldsymbol{{b}}\:\:{are}\:\:{prime}\:\:{numbers}\:. \\ $$$${Find}\:\:\boldsymbol{{a}}\:\:{and}\:\:\boldsymbol{{b}}\:. \\ $$

Question Number 31107 Answers: 0 Comments: 2

calculate ∫_(−∞) ^(+∞) (dx/((x^2 +x+1)^n )) with n>1.

$${calculate}\:\int_{−\infty} ^{+\infty} \:\:\:\:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} \:+{x}+\mathrm{1}\right)^{{n}} }\:\:{with}\:{n}>\mathrm{1}. \\ $$

Question Number 31106 Answers: 0 Comments: 0

prove that ∫_0 ^∞ e^(−x^2 ) =lim_(n→+∞) ∫_0 ^∞ (dx/((1+x^2 )^n )) . 2) prove that (1/(√π)) =lim_(n→∞) ((1.3.5....(2n−3))/(2.4.6....(2n−2))) (√n) (wallis formula).

$${prove}\:{that}\:\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{x}^{\mathrm{2}} } ={lim}_{{n}\rightarrow+\infty} \:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{{n}} }\:. \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\:\frac{\mathrm{1}}{\sqrt{\pi}}\:={lim}_{{n}\rightarrow\infty} \:\frac{\mathrm{1}.\mathrm{3}.\mathrm{5}....\left(\mathrm{2}{n}−\mathrm{3}\right)}{\mathrm{2}.\mathrm{4}.\mathrm{6}....\left(\mathrm{2}{n}−\mathrm{2}\right)}\:\sqrt{{n}} \\ $$$$\left({wallis}\:{formula}\right). \\ $$

Question Number 31105 Answers: 0 Comments: 1

prove that ∫_0 ^x e^(−t^2 ) dt =((√π)/2) −(e^(−x^2 ) /(√π)) ∫_0 ^∞ (e^(−x^2 t^2 ) /(1+t^2 )) dt with x>0

$${prove}\:{that}\:\int_{\mathrm{0}} ^{{x}} \:\:\:{e}^{−{t}^{\mathrm{2}} } {dt}\:=\frac{\sqrt{\pi}}{\mathrm{2}}\:−\frac{{e}^{−{x}^{\mathrm{2}} } }{\sqrt{\pi}}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−{x}^{\mathrm{2}} {t}^{\mathrm{2}} } }{\mathrm{1}+{t}^{\mathrm{2}} }\:{dt}\:{with}\:{x}>\mathrm{0} \\ $$

Question Number 31104 Answers: 0 Comments: 1

find ∫_(−∞) ^(+∞) e^(−(x^2 +2x−1)) dx .

$${find}\:\:\:\:\int_{−\infty} ^{+\infty} \:\:{e}^{−\left({x}^{\mathrm{2}} \:+\mathrm{2}{x}−\mathrm{1}\right)} {dx}\:. \\ $$

Question Number 31103 Answers: 0 Comments: 0

find ∫_0 ^∞ e^(−(x −(a/x))^2 ) dx with a≥0 .

$${find}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−\left({x}\:−\frac{{a}}{{x}}\right)^{\mathrm{2}} } {dx}\:\:{with}\:\:{a}\geqslant\mathrm{0}\:. \\ $$

Question Number 31102 Answers: 0 Comments: 2

find ∫_0 ^(+∞) ((lnx)/(x^2 +a^2 ))dx 2) find the value of ∫_0 ^∞ ((lnx)/((x^2 +a^2 )^3 )) .