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Question Number 144897    Answers: 0   Comments: 0

∫_0 ^(([x])/3) (8^x /2^([3x]) ) dx= ??? where [.] is the greatest integer function.

$$\underset{\mathrm{0}} {\overset{\frac{\left[{x}\right]}{\mathrm{3}}} {\int}}\frac{\mathrm{8}^{{x}} }{\mathrm{2}^{\left[\mathrm{3}{x}\right]} }\:{dx}=\:???\:\mathrm{where}\:\left[.\right]\:\mathrm{is}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{integer}\:\mathrm{function}. \\ $$

Question Number 144896    Answers: 1   Comments: 0

Evaluate ∫e^x (((1−x)/(1+x^2 )))^2 dx

$$\mathrm{Evaluate}\: \\ $$$$\:\int{e}^{{x}} \left(\frac{\mathrm{1}−{x}}{\mathrm{1}+{x}^{\mathrm{2}} }\right)^{\mathrm{2}} {dx}\: \\ $$

Question Number 144890    Answers: 1   Comments: 1

Question Number 144888    Answers: 2   Comments: 1

Question Number 144887    Answers: 1   Comments: 0

Question Number 144905    Answers: 0   Comments: 1

Question Number 144877    Answers: 1   Comments: 0

u+(√u)+(u)^(1/3) +(u)^(1/4) +(u)^(1/5) +... +∞=?

$$\:\:\mathrm{u}+\sqrt{\mathrm{u}}+\sqrt[{\mathrm{3}}]{\mathrm{u}}+\sqrt[{\mathrm{4}}]{\mathrm{u}}+\sqrt[{\mathrm{5}}]{\mathrm{u}}+...\:+\infty=? \\ $$$$ \\ $$

Question Number 144872    Answers: 0   Comments: 0

Question Number 144869    Answers: 1   Comments: 0

if 3^z = (1/(3^(5(√3)) ∙ 3^2 )) find z=?

$${if}\:\:\mathrm{3}^{\boldsymbol{{z}}} \:=\:\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{5}\sqrt{\mathrm{3}}} \:\centerdot\:\mathrm{3}^{\mathrm{2}} }\:\:{find}\:\:\boldsymbol{{z}}=? \\ $$

Question Number 144901    Answers: 0   Comments: 0

Let a,b > 0 and a+b+1 = 3ab. Prove that (a/(a^2 +1))+(b/(b^2 +1)) ≤ 1 ≤ (a^3 /(a^2 +1))+(b^3 /(b^2 +1)) Let a,b > 0, n ∈ Z^+ and a+b+1 = 3ab. Prove or disprove (a^(n−1) /(a^n +1))+(b^(n−1) /(b^n +1)) ≤ 1 ≤ (a^(n+1) /(a^n +1))+(b^(n+1) /(b^n +1))

$$\mathrm{Let}\:{a},{b}\:>\:\mathrm{0}\:\mathrm{and}\:{a}+{b}+\mathrm{1}\:=\:\mathrm{3}{ab}.\:\mathrm{Prove}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\frac{{a}}{{a}^{\mathrm{2}} +\mathrm{1}}+\frac{{b}}{{b}^{\mathrm{2}} +\mathrm{1}}\:\leqslant\:\mathrm{1}\:\leqslant\:\frac{{a}^{\mathrm{3}} }{{a}^{\mathrm{2}} +\mathrm{1}}+\frac{{b}^{\mathrm{3}} }{{b}^{\mathrm{2}} +\mathrm{1}} \\ $$$$ \\ $$$$\mathrm{Let}\:{a},{b}\:>\:\mathrm{0},\:{n}\:\in\:\mathbb{Z}^{+} \:\mathrm{and}\:{a}+{b}+\mathrm{1}\:=\:\mathrm{3}{ab}.\:\mathrm{Prove}\:\mathrm{or}\:\mathrm{disprove} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\frac{{a}^{{n}−\mathrm{1}} }{{a}^{{n}} +\mathrm{1}}+\frac{{b}^{{n}−\mathrm{1}} }{{b}^{{n}} +\mathrm{1}}\:\leqslant\:\mathrm{1}\:\leqslant\:\frac{{a}^{{n}+\mathrm{1}} }{{a}^{{n}} +\mathrm{1}}+\frac{{b}^{{n}+\mathrm{1}} }{{b}^{{n}} +\mathrm{1}} \\ $$$$ \\ $$

Question Number 144860    Answers: 0   Comments: 2

∣(x/(x-1))∣ + ∣x∣ = (x^2 /(∣x-1∣)) find x=?

$$\mid\frac{{x}}{{x}-\mathrm{1}}\mid\:+\:\mid{x}\mid\:=\:\frac{{x}^{\mathrm{2}} }{\mid{x}-\mathrm{1}\mid}\:\:\:{find}\:\:{x}=? \\ $$

Question Number 144858    Answers: 1   Comments: 0

Question Number 144857    Answers: 0   Comments: 0

Question Number 144849    Answers: 1   Comments: 0

∫_0 ^2 (1/(e^({x}^2 ) +1))dx {x} is fractional part of x

$$\int_{\mathrm{0}} ^{\mathrm{2}} \frac{\mathrm{1}}{{e}^{\left\{{x}\right\}^{\mathrm{2}} } +\mathrm{1}}{dx}\:\:\:\left\{{x}\right\}\:\:{is}\:{fractional}\:{part}\:{of}\:{x} \\ $$

Question Number 144876    Answers: 1   Comments: 0

∫_0 ^1 ((1/(sin x))−(1/x))dx=ln(2tan (1/2))

$$\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{sin}\:\mathrm{x}}−\frac{\mathrm{1}}{\mathrm{x}}\right)\mathrm{dx}=\mathrm{ln}\left(\mathrm{2tan}\:\frac{\mathrm{1}}{\mathrm{2}}\right) \\ $$

Question Number 144875    Answers: 1   Comments: 0

Question Number 144839    Answers: 3   Comments: 0

If x = (5)^(1/3) + 3 and y = 4 (3)^(1/3) Prove that: x - y < 0

$${If}\:\:{x}\:=\:\sqrt[{\mathrm{3}}]{\mathrm{5}}\:+\:\mathrm{3}\:\:{and}\:\:{y}\:=\:\mathrm{4}\:\sqrt[{\mathrm{3}}]{\mathrm{3}} \\ $$$${Prove}\:{that}:\:\:{x}\:-\:{y}\:<\:\mathrm{0} \\ $$

Question Number 144833    Answers: 1   Comments: 0

find the least value of α such that (4/(sin x))+(1/(1−sin x))=α has at least one solution in (0 (Π/2))

$$\mathrm{find}\:\mathrm{the}\:\mathrm{least}\:\mathrm{value}\:\mathrm{of}\:\alpha\:\mathrm{such}\:\mathrm{that}\:\frac{\mathrm{4}}{\mathrm{sin}\:\mathrm{x}}+\frac{\mathrm{1}}{\mathrm{1}−\mathrm{sin}\:\mathrm{x}}=\alpha\: \\ $$$$\mathrm{has}\:\mathrm{at}\:\mathrm{least}\:\mathrm{one}\:\mathrm{solution}\:\mathrm{in}\:\left(\mathrm{0}\:\frac{\Pi}{\mathrm{2}}\right) \\ $$

Question Number 144831    Answers: 2   Comments: 0

(3/(1∙2∙3)) + (5/(2∙3∙4)) + (7/(3∙4∙5)) + (9/(4∙5∙6)) + ... ∞=?

$$\frac{\mathrm{3}}{\mathrm{1}\centerdot\mathrm{2}\centerdot\mathrm{3}}\:+\:\frac{\mathrm{5}}{\mathrm{2}\centerdot\mathrm{3}\centerdot\mathrm{4}}\:+\:\frac{\mathrm{7}}{\mathrm{3}\centerdot\mathrm{4}\centerdot\mathrm{5}}\:+\:\frac{\mathrm{9}}{\mathrm{4}\centerdot\mathrm{5}\centerdot\mathrm{6}}\:+\:...\:\infty=? \\ $$

Question Number 144829    Answers: 2   Comments: 0

sin^3 x+cos^3 x=1−(1/2)sin^2 x x=?

$$\mathrm{sin}\:^{\mathrm{3}} {x}+\mathrm{cos}\:^{\mathrm{3}} {x}=\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{sin}\:^{\mathrm{2}} {x} \\ $$$${x}=? \\ $$

Question Number 144828    Answers: 2   Comments: 0

2sin 17x+(√3) cos 5x+sin 5x=0 x=?

$$\mathrm{2sin}\:\mathrm{17}{x}+\sqrt{\mathrm{3}}\:\mathrm{cos}\:\mathrm{5}{x}+\mathrm{sin}\:\mathrm{5}{x}=\mathrm{0} \\ $$$${x}=? \\ $$

Question Number 144826    Answers: 1   Comments: 0

Question Number 144825    Answers: 1   Comments: 0

Σ_(k=0) ^∞ (k/(k^4 + 4)) = ?

$$\underset{\boldsymbol{{k}}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{{k}}{{k}^{\mathrm{4}} \:+\:\mathrm{4}}\:=\:? \\ $$

Question Number 144823    Answers: 1   Comments: 0

Let a,b > 0 and a+b+1 = 3ab. Prove that ((a+1)/(b+1))+((b+1)/(a+1)) ≤ a+b

$$\mathrm{Let}\:{a},{b}\:>\:\mathrm{0}\:\mathrm{and}\:{a}+{b}+\mathrm{1}\:=\:\mathrm{3}{ab}.\:\mathrm{Prove}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{{a}+\mathrm{1}}{{b}+\mathrm{1}}+\frac{{b}+\mathrm{1}}{{a}+\mathrm{1}}\:\leqslant\:{a}+{b} \\ $$

Question Number 144822    Answers: 1   Comments: 0

sin^3 xcos x−cos^3 xsin x=(1/4) x=?

$$\mathrm{sin}\:^{\mathrm{3}} {x}\mathrm{cos}\:{x}−\mathrm{cos}\:^{\mathrm{3}} {x}\mathrm{sin}\:{x}=\frac{\mathrm{1}}{\mathrm{4}} \\ $$$${x}=? \\ $$

Question Number 144821    Answers: 0   Comments: 1

tan 193=k cos 167=?

$$\mathrm{tan}\:\mathrm{193}={k} \\ $$$$\mathrm{cos}\:\mathrm{167}=? \\ $$

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