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x and y are reals(or complex) let put x^((0)) =1 ,x^((1)) =x x^((2)) =x(x−1).....x^((n)) =x(x−1)(x−2)...(x−n+1)prove that (x+y)^((n)) =Σ_(k=0) ^n C_n ^k x^((n−k)) y^((k))

$${x}\:{and}\:{y}\:{are}\:{reals}\left({or}\:{complex}\right)\:{let}\:{put}\:{x}^{\left(\mathrm{0}\right)} =\mathrm{1}\:,{x}^{\left(\mathrm{1}\right)} ={x} \\ $$$${x}^{\left(\mathrm{2}\right)} ={x}\left({x}−\mathrm{1}\right).....{x}^{\left({n}\right)} ={x}\left({x}−\mathrm{1}\right)\left({x}−\mathrm{2}\right)...\left({x}−{n}+\mathrm{1}\right){prove}\:{that} \\ $$$$\left({x}+{y}\right)^{\left({n}\right)} =\sum_{{k}=\mathrm{0}} ^{{n}} \:{C}_{{n}} ^{{k}} \:\:{x}^{\left({n}−{k}\right)} {y}^{\left({k}\right)} \\ $$

Question Number 73021 Answers: 0 Comments: 0

Question Number 73017 Answers: 0 Comments: 1

find lim_(x→+∞) x(√(x^2 + 1))

$${find}\: \\ $$$$\:\underset{{x}\rightarrow+\infty} {\mathrm{lim}}\:\:{x}\sqrt{{x}^{\mathrm{2}} \:+\:\mathrm{1}}\: \\ $$

Question Number 73084 Answers: 1 Comments: 0

Question Number 72998 Answers: 0 Comments: 0

The acute angle of the rectangle trapezius is equal to α=90°arcsin0.1 The bases measure 10 and 30. Calculate the area of the trapezius.

$${The}\:{acute}\:{angle}\:{of}\:{the}\:{rectangle}\:{trapezius}\:{is}\:{equal}\:{to}\:\alpha=\mathrm{90}°{arcsin}\mathrm{0}.\mathrm{1} \\ $$$${The}\:{bases}\:{measure}\:\mathrm{10}\:{and}\:\mathrm{30}.\:{Calculate}\:{the}\:{area}\:{of}\:{the}\:{trapezius}. \\ $$

Question Number 72997 Answers: 1 Comments: 0

The area of the equilateral triangle is equal to (((√(16))(√8))/(3(√π))) Calculate the area of the circle inscribed in the triangle.

$${The}\:{area}\:{of}\:{the}\:{equilateral}\:{triangle}\:{is}\:{equal}\:{to}\:\frac{\sqrt{\mathrm{16}}\sqrt{\mathrm{8}}}{\mathrm{3}\sqrt{\pi}} \\ $$$${Calculate}\:{the}\:{area}\:{of}\:{the}\:{circle}\:{inscribed}\:{in}\:{the}\:{triangle}. \\ $$$$\: \\ $$

Question Number 72990 Answers: 1 Comments: 1

calculate lim_(x→0) ((arctan(e^x )−(π/4))/x^2 )

$${calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\:\:\frac{{arctan}\left({e}^{{x}} \right)−\frac{\pi}{\mathrm{4}}}{{x}^{\mathrm{2}} } \\ $$

Question Number 72988 Answers: 1 Comments: 1

calculate f(x)=∫_0 ^∞ (e^(−xt^2 ) /(4+t^2 ))dt with x>0

$${calculate}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{e}^{−{xt}^{\mathrm{2}} } }{\mathrm{4}+{t}^{\mathrm{2}} }{dt}\:\:\:{with}\:{x}>\mathrm{0} \\ $$

Question Number 72986 Answers: 0 Comments: 0

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

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

Question Number 72965 Answers: 1 Comments: 1

Prove that this equation is true: Π_(x=1) ^(n−1) (2x+1)=(((2x−1)!)/((2)^(x−1) (x−1)!))

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{this}\:\mathrm{equation}\:\mathrm{is}\:\mathrm{true}: \\ $$$$\underset{{x}=\mathrm{1}} {\overset{{n}−\mathrm{1}} {\prod}}\left(\mathrm{2}{x}+\mathrm{1}\right)=\frac{\left(\mathrm{2}{x}−\mathrm{1}\right)!}{\left(\mathrm{2}\right)^{{x}−\mathrm{1}} \left({x}−\mathrm{1}\right)!} \\ $$

Question Number 72952 Answers: 2 Comments: 4

Question Number 72931 Answers: 2 Comments: 0

Find the numeric value of ((2−sin^2 (α))/(cos^2 (α)−tan^2 (α)))

$${Find}\:{the}\:{numeric}\:{value}\:{of} \\ $$$$\frac{\mathrm{2}−\mathrm{sin}\:^{\mathrm{2}} \left(\alpha\right)}{\mathrm{cos}\:^{\mathrm{2}} \left(\alpha\right)−\mathrm{tan}\:^{\mathrm{2}} \left(\alpha\right)} \\ $$

Question Number 72930 Answers: 1 Comments: 0

Question Number 73012 Answers: 4 Comments: 0

∫(√(tanx)) dx

$$\int\sqrt{{tan}\mathrm{x}}\:{d}\mathrm{x} \\ $$

Question Number 72912 Answers: 1 Comments: 0

calculate S_p = Σ_(n=0) ^∞ (((−1)^n )/(n+p))

$${calculate}\:{S}_{{p}} =\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}+{p}} \\ $$

Question Number 72908 Answers: 2 Comments: 15

find ∫_0 ^π (dθ/(x^2 −2x cosθ +1)) with x real.

$${find}\:\int_{\mathrm{0}} ^{\pi} \:\:\frac{{d}\theta}{{x}^{\mathrm{2}} −\mathrm{2}{x}\:{cos}\theta\:+\mathrm{1}}\:\:{with}\:{x}\:{real}. \\ $$

Question Number 72900 Answers: 1 Comments: 0

prove that −∣a∣≤a≤∣a∣ a is a real number

$${prove}\:{that}\: \\ $$$$ \\ $$$$−\mid{a}\mid\leqslant{a}\leqslant\mid{a}\mid \\ $$$$ \\ $$$${a}\:{is}\:{a}\:{real}\:{number} \\ $$

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