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Question Number 146102 Answers: 1 Comments: 0

prove by mathmatical indiction 5+7+9+.....+(4n+1)=2n^2 +3n

$${prove}\:{by}\:{mathmatical}\:{indiction}\: \\ $$$$\mathrm{5}+\mathrm{7}+\mathrm{9}+.....+\left(\mathrm{4}{n}+\mathrm{1}\right)=\mathrm{2}{n}^{\mathrm{2}} +\mathrm{3}{n} \\ $$

Question Number 146100 Answers: 1 Comments: 1

Question Number 146096 Answers: 1 Comments: 0

Question Number 146090 Answers: 2 Comments: 0

∫_0 ^∞ ((sinh(at)sinh(bt))/(sinh(ct)e^(tz) ))dt= ((ab)/(c(z^2 +c^2 −a^2 −b^2 +K_(k=1) ^∞ ((−4k^2 (k^2 c^2 −a^2 )(k^2 c^2 −b^2 ))/((2k+1)(z^2 +(2k^2 +2k+1)c^2 −a^2 −b^2 ))))))

$$\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{sinh}\left(\mathrm{at}\right)\mathrm{sinh}\left(\mathrm{bt}\right)}{\mathrm{sinh}\left(\mathrm{ct}\right)\mathrm{e}^{\mathrm{tz}} }\mathrm{dt}= \\ $$$$\frac{\mathrm{ab}}{\mathrm{c}\left(\mathrm{z}^{\mathrm{2}} +\mathrm{c}^{\mathrm{2}} −\mathrm{a}^{\mathrm{2}} −\mathrm{b}^{\mathrm{2}} +\underset{\mathrm{k}=\mathrm{1}} {\overset{\infty} {\mathrm{K}}}\frac{−\mathrm{4k}^{\mathrm{2}} \left(\mathrm{k}^{\mathrm{2}} \mathrm{c}^{\mathrm{2}} −\mathrm{a}^{\mathrm{2}} \right)\left(\mathrm{k}^{\mathrm{2}} \mathrm{c}^{\mathrm{2}} −\mathrm{b}^{\mathrm{2}} \right)}{\left(\mathrm{2k}+\mathrm{1}\right)\left(\mathrm{z}^{\mathrm{2}} +\left(\mathrm{2k}^{\mathrm{2}} +\mathrm{2k}+\mathrm{1}\right)\mathrm{c}^{\mathrm{2}} −\mathrm{a}^{\mathrm{2}} −\mathrm{b}^{\mathrm{2}} \right)}\right)} \\ $$

Question Number 146091 Answers: 0 Comments: 0

Question Number 146087 Answers: 1 Comments: 0

1)find U_n =∫_0 ^1 x^n e^(−2x) dx 2)nature of Σ U_n ?

$$\left.\mathrm{1}\right)\mathrm{find}\:\mathrm{U}_{\mathrm{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \:\mathrm{x}^{\mathrm{n}} \:\mathrm{e}^{−\mathrm{2x}} \:\mathrm{dx} \\ $$$$\left.\mathrm{2}\right)\mathrm{nature}\:\mathrm{of}\:\Sigma\:\mathrm{U}_{\mathrm{n}} ? \\ $$

Question Number 146085 Answers: 0 Comments: 1

f(x,y)=x−(√(x+2y)) 1)condition on x and y to have f symetric 2) find (∂f/∂x) ,(∂f/∂y) ,(∂^2 f/(∂x∂y)) ,(∂^2 f/(∂y∂x)) 3) find (∂^2 f/∂^2 x) and (∂^2 f/∂^2 y)

$$\mathrm{f}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{x}−\sqrt{\mathrm{x}+\mathrm{2y}} \\ $$$$\left.\mathrm{1}\right)\mathrm{condition}\:\mathrm{on}\:\mathrm{x}\:\mathrm{and}\:\mathrm{y}\:\mathrm{to}\:\mathrm{have}\:\mathrm{f}\:\mathrm{symetric} \\ $$$$\left.\mathrm{2}\right)\:\mathrm{find}\:\frac{\partial\mathrm{f}}{\partial\mathrm{x}}\:,\frac{\partial\mathrm{f}}{\partial\mathrm{y}}\:,\frac{\partial^{\mathrm{2}} \mathrm{f}}{\partial\mathrm{x}\partial\mathrm{y}}\:,\frac{\partial^{\mathrm{2}} \mathrm{f}}{\partial\mathrm{y}\partial\mathrm{x}} \\ $$$$\left.\mathrm{3}\right)\:\mathrm{find}\:\frac{\partial^{\mathrm{2}} \mathrm{f}}{\partial^{\mathrm{2}} \mathrm{x}}\:\mathrm{and}\:\frac{\partial^{\mathrm{2}} \mathrm{f}}{\partial^{\mathrm{2}} \mathrm{y}} \\ $$

Question Number 146083 Answers: 1 Comments: 0

F(x)=x^n −e^(inα) 1) roots of F(x)? 2) factorize F(x) inside C[x]

$$\mathrm{F}\left(\mathrm{x}\right)=\mathrm{x}^{\mathrm{n}} \:−\mathrm{e}^{\mathrm{in}\alpha} \\ $$$$\left.\mathrm{1}\right)\:\mathrm{roots}\:\mathrm{of}\:\mathrm{F}\left(\mathrm{x}\right)? \\ $$$$\left.\mathrm{2}\right)\:\mathrm{factorize}\:\mathrm{F}\left(\mathrm{x}\right)\:\mathrm{inside}\:\mathrm{C}\left[\mathrm{x}\right] \\ $$

Question Number 146082 Answers: 0 Comments: 0

p(x)=(x^2 −x+1)^n −(x^2 +x+1)^n 1) roots of p(x)? 2) factorize p(x) inside C[x]

$$\mathrm{p}\left(\mathrm{x}\right)=\left(\mathrm{x}^{\mathrm{2}} −\mathrm{x}+\mathrm{1}\right)^{\mathrm{n}} −\left(\mathrm{x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}\right)^{\mathrm{n}} \\ $$$$\left.\mathrm{1}\right)\:\mathrm{roots}\:\mathrm{of}\:\mathrm{p}\left(\mathrm{x}\right)? \\ $$$$\left.\mathrm{2}\right)\:\mathrm{factorize}\:\mathrm{p}\left(\mathrm{x}\right)\:\mathrm{inside}\:\mathrm{C}\left[\mathrm{x}\right] \\ $$

Question Number 146076 Answers: 1 Comments: 0

if f(x)=∫(3x^2 −2)dx and f(2)=9 then f(−2)= ?

$${if}\:{f}\left({x}\right)=\int\left(\mathrm{3}{x}^{\mathrm{2}} −\mathrm{2}\right){dx}\:{and}\:{f}\left(\mathrm{2}\right)=\mathrm{9}\:{then} \\ $$$${f}\left(−\mathrm{2}\right)=\:? \\ $$

Question Number 146073 Answers: 0 Comments: 0

Let K be nonempty corps , K^∗ =K−{0_K } Prove that 1) Π_(x∈K^∗ ) x = −1 2)Deduce that p is prime ⇔ (p−1)!≡−1[p]