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AllQuestion and Answers: Page 751

Question Number 143094 Answers: 1 Comments: 0

Question Number 143090 Answers: 0 Comments: 0

Question Number 143087 Answers: 2 Comments: 0

Question Number 143086 Answers: 2 Comments: 0

Evaluate :: Ω:=∫_0 ^( (π/4)) ((ln(tan(x)).sin^π^e (2x))/((sin^π^e (x)+cos^π^e (x))^2 ))dx

$$ \\ $$$$\:\:{Evaluate}\::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\Omega:=\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{4}}} \frac{{ln}\left({tan}\left({x}\right)\right).{sin}^{\pi^{{e}} } \left(\mathrm{2}{x}\right)}{\left({sin}^{\pi^{{e}} } \left({x}\right)+{cos}^{\pi^{{e}} } \left({x}\right)\right)^{\mathrm{2}} }{dx} \\ $$$$ \\ $$

Question Number 143085 Answers: 0 Comments: 0

φ(n^4 +1)=8n φ:Euler totient function Solve for n∈N

$$\phi\left({n}^{\mathrm{4}} +\mathrm{1}\right)=\mathrm{8}{n}\:\:\:\:\:\:\phi:{Euler}\:{totient}\:{function} \\ $$$${Solve}\:{for}\:{n}\in\mathbb{N} \\ $$

Question Number 143083 Answers: 1 Comments: 0

calculate Ψ(a,b)=∫_0 ^∞ (e^(−ax^2 ) /((x^2 +b^2 )^2 ))dx with a>0 and b>0

$${calculate}\:\Psi\left({a},{b}\right)=\int_{\mathrm{0}} ^{\infty} \:\frac{{e}^{−{ax}^{\mathrm{2}} } }{\left({x}^{\mathrm{2}} \:+{b}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx} \\ $$$${with}\:{a}>\mathrm{0}\:{and}\:{b}>\mathrm{0} \\ $$

Question Number 143082 Answers: 2 Comments: 0

calculate f(a,b)=∫_0 ^∞ (e^(−ax^2 ) /(x^2 +b^2 ))dx with a>0 and b>0

$${calculate}\:{f}\left({a},{b}\right)=\int_{\mathrm{0}} ^{\infty} \:\frac{{e}^{−{ax}^{\mathrm{2}} } }{{x}^{\mathrm{2}} \:+{b}^{\mathrm{2}} }{dx} \\ $$$${with}\:{a}>\mathrm{0}\:{and}\:{b}>\mathrm{0} \\ $$

Question Number 143081 Answers: 2 Comments: 0

calculate ∫_0 ^∞ xe^(−x^2 ) arctanx dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} {xe}^{−{x}^{\mathrm{2}} } {arctanx}\:{dx} \\ $$

Question Number 143080 Answers: 2 Comments: 0

calculate ∫_0 ^∞ ((arctan(x^2 ))/(1+x^2 ))dx

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

Question Number 143077 Answers: 1 Comments: 0

sin^5 x + cos^5 x = 2 − sin^4 x

$${sin}^{\mathrm{5}} {x}\:+\:{cos}^{\mathrm{5}} {x}\:=\:\mathrm{2}\:−\:{sin}^{\mathrm{4}} {x} \\ $$

Question Number 143072 Answers: 1 Comments: 0

Question Number 143071 Answers: 2 Comments: 0

∫_0 ^(π/4) ((8dx)/(tgx+1))

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \frac{\mathrm{8}{dx}}{{tgx}+\mathrm{1}} \\ $$

Question Number 143064 Answers: 4 Comments: 0

lim_(x→0) ((((1+x^2 ))^(1/3) −((1−2x))^(1/4) )/(x+x^2 )) =? lim_(x→1) ((((7+x^2 ))^(1/3) −(√(3+x^2 )))/(x−1)) =?

$$\:\:\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt[{\mathrm{3}}]{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\:−\sqrt[{\mathrm{4}}]{\mathrm{1}−\mathrm{2x}}}{\mathrm{x}+\mathrm{x}^{\mathrm{2}} }\:=? \\ $$$$\:\:\:\:\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\frac{\sqrt[{\mathrm{3}}]{\mathrm{7}+\mathrm{x}^{\mathrm{2}} }−\sqrt{\mathrm{3}+\mathrm{x}^{\mathrm{2}} }}{\mathrm{x}−\mathrm{1}}\:=? \\ $$

Question Number 143063 Answers: 0 Comments: 0

cos((𝛑/(2n+1)))cos(((2𝛑)/(2n+1)))cos(((3𝛑)/(2n+1))).....cos(((n𝛑)/(2n+1)))=(1/2^n ) prove

$$\boldsymbol{\mathrm{cos}}\left(\frac{\boldsymbol{\pi}}{\mathrm{2n}+\mathrm{1}}\right)\boldsymbol{\mathrm{cos}}\left(\frac{\mathrm{2}\boldsymbol{\pi}}{\mathrm{2n}+\mathrm{1}}\right)\boldsymbol{\mathrm{cos}}\left(\frac{\mathrm{3}\boldsymbol{\pi}}{\mathrm{2n}+\mathrm{1}}\right).....\boldsymbol{\mathrm{cos}}\left(\frac{\boldsymbol{\mathrm{n}\pi}}{\mathrm{2}\boldsymbol{\mathrm{n}}+\mathrm{1}}\right)=\frac{\mathrm{1}}{\mathrm{2}^{\boldsymbol{\mathrm{n}}} } \\ $$$$\boldsymbol{\mathrm{prove}} \\ $$

Question Number 143057 Answers: 1 Comments: 0

cos(𝛂)×cos(2α)×cos(4α)×....×cos(2^n 𝛂)=((sin(2^(n+1) 𝛂))/(2^(n+1) sin(α))) prove

$$\mathrm{cos}\left(\boldsymbol{\alpha}\right)×\mathrm{cos}\left(\mathrm{2}\alpha\right)×\mathrm{cos}\left(\mathrm{4}\alpha\right)×....×\mathrm{cos}\left(\mathrm{2}^{\mathrm{n}} \boldsymbol{\alpha}\right)=\frac{\boldsymbol{\mathrm{sin}}\left(\mathrm{2}^{\boldsymbol{\mathrm{n}}+\mathrm{1}} \boldsymbol{\alpha}\right)}{\mathrm{2}^{\mathrm{n}+\mathrm{1}} \mathrm{sin}\left(\alpha\right)} \\ $$$$\boldsymbol{\mathrm{prove}} \\ $$

Question Number 143048 Answers: 1 Comments: 0

Question Number 143047 Answers: 2 Comments: 0

Question Number 143046 Answers: 1 Comments: 0

Question Number 143045 Answers: 1 Comments: 0

Question Number 143043 Answers: 1 Comments: 0

Question Number 143042 Answers: 1 Comments: 0

Question Number 143041 Answers: 0 Comments: 2

If f(x)=x^2 +4x+2 then the value of (1−(2/(f(1))))(1−(2/(f(2))))(1−(2/(f(3))))...(1−(2/(f(2021))))=?

$${If}\:{f}\left({x}\right)={x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{2}\:{then}\:{the}\:{value} \\ $$$${of}\:\left(\mathrm{1}−\frac{\mathrm{2}}{{f}\left(\mathrm{1}\right)}\right)\left(\mathrm{1}−\frac{\mathrm{2}}{{f}\left(\mathrm{2}\right)}\right)\left(\mathrm{1}−\frac{\mathrm{2}}{{f}\left(\mathrm{3}\right)}\right)...\left(\mathrm{1}−\frac{\mathrm{2}}{{f}\left(\mathrm{2021}\right)}\right)=? \\ $$

Question Number 143039 Answers: 0 Comments: 0

cos^(−1) (((x^2 −1)/(x^2 +1)))+(1/2)tan^(−1) (((2x)/(1−x^2 )))=((2π)/3) x=?

$$\:\mathrm{cos}^{−\mathrm{1}} \left(\frac{{x}^{\mathrm{2}} −\mathrm{1}}{{x}^{\mathrm{2}} +\mathrm{1}}\right)+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{2}{x}}{\mathrm{1}−{x}^{\mathrm{2}} }\right)=\frac{\mathrm{2}\pi}{\mathrm{3}} \\ $$$${x}=? \\ $$

Question Number 143051 Answers: 1 Comments: 0

_(∗∗∗∗∗) :: Lobachevsky Integral ::_(∗∗∗∗∗) 𝛗:=∫_0 ^( ∞) ((sin^2 ( tan(x)))/x^( 2) )dx=^? (π/2) ..........

$$\:\:\:\:\:\:\:_{\ast\ast\ast\ast\ast} ::\:\:{Lobachevsky}\:{Integral}\:::_{\ast\ast\ast\ast\ast} \\ $$$$\:\:\:\:\:\:\:\:\:\boldsymbol{\phi}:=\int_{\mathrm{0}} ^{\:\infty} \frac{\mathrm{s}{in}^{\mathrm{2}} \left(\:{tan}\left({x}\right)\right)}{{x}^{\:\mathrm{2}} }{dx}\overset{?} {=}\frac{\pi}{\mathrm{2}} \\ $$$$\:\:\:\:.......... \\ $$

Question Number 143036 Answers: 0 Comments: 0

x^(3/2) +x^(1/2) +(x−c)(((3x+1)/(3+x)))^(3/2) =0

$$\:{x}^{\mathrm{3}/\mathrm{2}} +{x}^{\mathrm{1}/\mathrm{2}} +\left({x}−{c}\right)\left(\frac{\mathrm{3}{x}+\mathrm{1}}{\mathrm{3}+{x}}\right)^{\mathrm{3}/\mathrm{2}} =\mathrm{0} \\ $$

Question Number 143032 Answers: 1 Comments: 0

∫ sin^(−5) x dx =?

$$\:\:\:\:\:\int\:\mathrm{sin}^{−\mathrm{5}} {x}\:{dx}\:=? \\ $$

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