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Question Number 155510 Answers: 1 Comments: 0

Question Number 155506 Answers: 1 Comments: 0

Question Number 155500 Answers: 2 Comments: 0

Question Number 155497 Answers: 1 Comments: 0

Question Number 155496 Answers: 1 Comments: 0

Given I_n =∫_(nπ) ^((n+1)π) e^(−x) sinx dx , n∈N. 1. Find a relation between I_(n+1) and I_n .

$${Given}\:{I}_{{n}} =\underset{{n}\pi} {\overset{\left({n}+\mathrm{1}\right)\pi} {\int}}{e}^{−{x}} {sinx}\:{dx}\:,\:{n}\in\mathbb{N}. \\ $$$$\mathrm{1}.\:{Find}\:{a}\:{relation}\:{between}\:{I}_{{n}+\mathrm{1}} {and}\:{I}_{{n}} . \\ $$

Question Number 155495 Answers: 1 Comments: 0

if a;b;c;d∈R verify a+2b+3c+4d=6 then find min(a^2 +b^2 +c^2 +d^2 )

$$\mathrm{if}\:\:\mathrm{a};\mathrm{b};\mathrm{c};\mathrm{d}\in\mathbb{R}\:\:\mathrm{verify}\:\:\mathrm{a}+\mathrm{2b}+\mathrm{3c}+\mathrm{4d}=\mathrm{6} \\ $$$$\mathrm{then}\:\mathrm{find}\:\:\boldsymbol{\mathrm{min}}\left(\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} +\mathrm{c}^{\mathrm{2}} +\mathrm{d}^{\mathrm{2}} \right) \\ $$

Question Number 155488 Answers: 1 Comments: 0

Question Number 155481 Answers: 1 Comments: 0

if a;b;c∈[1;∞) then prove that a^(1/a) ; b^(1/b) ; c^(1/c) are the sides of a triangle.

$$\mathrm{if}\:\:\mathrm{a};\mathrm{b};\mathrm{c}\in\left[\mathrm{1};\infty\right) \\ $$$$\mathrm{then}\:\mathrm{prove}\:\mathrm{that}\:\:\mathrm{a}^{\frac{\mathrm{1}}{\boldsymbol{\mathrm{a}}}} \:;\:\mathrm{b}^{\frac{\mathrm{1}}{\boldsymbol{\mathrm{b}}}} \:;\:\mathrm{c}^{\frac{\mathrm{1}}{\boldsymbol{\mathrm{c}}}} \\ $$$$\mathrm{are}\:\mathrm{the}\:\mathrm{sides}\:\mathrm{of}\:\mathrm{a}\:\mathrm{triangle}. \\ $$

Question Number 155480 Answers: 1 Comments: 0

Find the positive integer solution of the equation: x^3 + y^3 = 911(xy + 49)

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{positive}\:\mathrm{integer}\:\mathrm{solution} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{equation}: \\ $$$$\mathrm{x}^{\mathrm{3}} \:+\:\mathrm{y}^{\mathrm{3}} \:=\:\mathrm{911}\left(\mathrm{xy}\:+\:\mathrm{49}\right) \\ $$

Question Number 155479 Answers: 1 Comments: 5

Find: 𝛀 =∫_( 0) ^( ∞) (((√x) ln(x))/(x^2 + 1)) dx = ?

$$\mathrm{Find}:\:\:\boldsymbol{\Omega}\:=\underset{\:\mathrm{0}} {\overset{\:\infty} {\int}}\:\frac{\sqrt{\mathrm{x}}\:\mathrm{ln}\left(\mathrm{x}\right)}{\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{1}}\:\mathrm{dx}\:=\:? \\ $$

Question Number 155478 Answers: 1 Comments: 0

Π_(k = 1 ) ^n (1 + (1/k))^k = V_n find V_n

$$\underset{{k}\:=\:\mathrm{1}\:} {\overset{{n}} {\prod}}\left(\mathrm{1}\:+\:\frac{\mathrm{1}}{{k}}\right)^{{k}} \:=\:{V}_{{n}} \\ $$$${find}\:{V}_{{n}} \\ $$

Question Number 155477 Answers: 1 Comments: 0

Question Number 155512 Answers: 1 Comments: 0

𝛀 =∫_( 0) ^( 1) ((tan^(-1) x - tan^(-1) ((1/x)))/(1 + x^2 )) ∙ ((1+x)/(1-x)) dx = ?

$$\boldsymbol{\Omega}\:=\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:\frac{\mathrm{tan}^{-\mathrm{1}} \boldsymbol{\mathrm{x}}\:-\:\mathrm{tan}^{-\mathrm{1}} \left(\frac{\mathrm{1}}{\boldsymbol{\mathrm{x}}}\right)}{\mathrm{1}\:+\:\mathrm{x}^{\mathrm{2}} }\:\centerdot\:\frac{\mathrm{1}+\mathrm{x}}{\mathrm{1}-\mathrm{x}}\:\mathrm{dx}\:=\:? \\ $$

Question Number 155466 Answers: 2 Comments: 2

tan^2 ((π/(16)))+tan^2 (((2π)/(16)))+…+tan^2 (((7π)/(16)))=?

$$\:\mathrm{tan}\:^{\mathrm{2}} \left(\frac{\pi}{\mathrm{16}}\right)+\mathrm{tan}\:^{\mathrm{2}} \left(\frac{\mathrm{2}\pi}{\mathrm{16}}\right)+\ldots+\mathrm{tan}\:^{\mathrm{2}} \left(\frac{\mathrm{7}\pi}{\mathrm{16}}\right)=? \\ $$

Question Number 155465 Answers: 1 Comments: 0

(1^2 /(1×3))+(2^2 /(3×5))+(3^2 /(5×7))+...+(n^2 /((2n−1)(2n+1)))=

$$\frac{\mathrm{1}^{\mathrm{2}} }{\mathrm{1}×\mathrm{3}}+\frac{\mathrm{2}^{\mathrm{2}} }{\mathrm{3}×\mathrm{5}}+\frac{\mathrm{3}^{\mathrm{2}} }{\mathrm{5}×\mathrm{7}}+...+\frac{\mathrm{n}^{\mathrm{2}} }{\left(\mathrm{2n}−\mathrm{1}\right)\left(\mathrm{2n}+\mathrm{1}\right)}= \\ $$

Question Number 155462 Answers: 0 Comments: 0

Question Number 155461 Answers: 0 Comments: 0

Question Number 155450 Answers: 2 Comments: 0

Question Number 155438 Answers: 0 Comments: 0

Question Number 155436 Answers: 0 Comments: 0

S = Σ_(m=0) ^∞ (1/(Σ_(n=0) ^m 10^n + m))

$$\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:{S}\:=\:\underset{{m}=\mathrm{0}} {\overset{\infty} {\sum}}\:\:\frac{\mathrm{1}}{\underset{{n}=\mathrm{0}} {\overset{{m}} {\sum}}\:\mathrm{10}^{{n}} +\:{m}} \\ $$$$\: \\ $$

Question Number 155428 Answers: 2 Comments: 0

simplify t=(((26)/(3(√3)))+5)^(1/3) −(((26)/(3(√3)))−5)^(1/3)

$${simplify} \\ $$$${t}=\left(\frac{\mathrm{26}}{\mathrm{3}\sqrt{\mathrm{3}}}+\mathrm{5}\right)^{\mathrm{1}/\mathrm{3}} −\left(\frac{\mathrm{26}}{\mathrm{3}\sqrt{\mathrm{3}}}−\mathrm{5}\right)^{\mathrm{1}/\mathrm{3}} \\ $$

Question Number 155426 Answers: 1 Comments: 2

Question Number 155420 Answers: 1 Comments: 0

(1/(42))+(1/(72))+(1/(110))+…=?

$$\frac{\mathrm{1}}{\mathrm{42}}+\frac{\mathrm{1}}{\mathrm{72}}+\frac{\mathrm{1}}{\mathrm{110}}+\ldots=? \\ $$

Question Number 155400 Answers: 0 Comments: 1

x^2 +x+5xy+6y^2 +2y−2=0

$${x}^{\mathrm{2}} +{x}+\mathrm{5}{xy}+\mathrm{6}{y}^{\mathrm{2}} +\mathrm{2}{y}−\mathrm{2}=\mathrm{0} \\ $$

Question Number 155399 Answers: 2 Comments: 0

Question Number 155393 Answers: 1 Comments: 0

∣∣x−4∣−4∣ ≥ 4

$$\:\:\mid\mid\mathrm{x}−\mathrm{4}\mid−\mathrm{4}\mid\:\geqslant\:\mathrm{4}\: \\ $$

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