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Question Number 213838 Answers: 1 Comments: 1

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Find: lim_(x→0) (((sinx)/x))^((sinx)/(x − sinx)) = ?

$$\mathrm{Find}:\:\:\:\underset{\boldsymbol{\mathrm{x}}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\frac{\mathrm{sinx}}{\mathrm{x}}\right)^{\frac{\mathrm{sinx}}{\mathrm{x}\:−\:\mathrm{sinx}}} \:\:=\:\:? \\ $$

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Question Number 213791 Answers: 2 Comments: 2

If x − (x)^(1/3) − (4/( (x)^(1/3) )) = 10 Find (x)^(1/3) − (1/( (x)^(1/3) )) + 3 = ?

$$\mathrm{If}\:\:\:\mathrm{x}\:−\:\sqrt[{\mathrm{3}}]{\mathrm{x}}\:−\:\frac{\mathrm{4}}{\:\sqrt[{\mathrm{3}}]{\mathrm{x}}}\:\:=\:\:\mathrm{10} \\ $$$$\mathrm{Find}\:\:\:\sqrt[{\mathrm{3}}]{\mathrm{x}}\:−\:\frac{\mathrm{1}}{\:\sqrt[{\mathrm{3}}]{\mathrm{x}}}\:\:+\:\:\mathrm{3}\:\:=\:\:? \\ $$

Question Number 213790 Answers: 0 Comments: 2

So Weird...... ∫_0 ^( ∞) J_ν (t)e^(−st) dt=(((s+(√(s^2 +1)))^(−ν) )/( (√(s^2 +1)))) J_(−ν) (t)=(−1)^ν J_ν (t) ∫_0 ^( ∞) J_(−ν) (t)e^(−st) dt=(((−1)^ν (s+(√(s^2 +1)))^(−ν) )/( (√(s^2 +1)))) is true But ∫_0 ^( ∞) J_(−ν) (t)e^(−st) dt is not (((s+(√(s^2 +1)))^ν )/( (√(s^2 +1)))) why....? can you explain why Blue equation is not true....

$$\mathrm{So}\:\mathrm{Weird}...... \\ $$$$\int_{\mathrm{0}} ^{\:\infty} {J}_{\nu} \left({t}\right){e}^{−{st}} \mathrm{d}{t}=\frac{\left({s}+\sqrt{{s}^{\mathrm{2}} +\mathrm{1}}\right)^{−\nu} }{\:\sqrt{{s}^{\mathrm{2}} +\mathrm{1}}}\: \\ $$$${J}_{−\nu} \left({t}\right)=\left(−\mathrm{1}\right)^{\nu} {J}_{\nu} \left({t}\right)\:\: \\ $$$$\int_{\mathrm{0}} ^{\:\infty} \:{J}_{−\nu} \left({t}\right){e}^{−{st}} \mathrm{d}{t}=\frac{\left(−\mathrm{1}\right)^{\nu} \left({s}+\sqrt{{s}^{\mathrm{2}} +\mathrm{1}}\right)^{−\nu} }{\:\sqrt{{s}^{\mathrm{2}} +\mathrm{1}}}\:\mathrm{is}\:\mathrm{true} \\ $$$$\mathrm{But}\:\int_{\mathrm{0}} ^{\:\infty} \:{J}_{−\nu} \left({t}\right){e}^{−{st}} \mathrm{d}{t}\:\mathrm{is}\:\mathrm{not}\:\frac{\left({s}+\sqrt{{s}^{\mathrm{2}} +\mathrm{1}}\right)^{\nu} }{\:\sqrt{{s}^{\mathrm{2}} +\mathrm{1}}} \\ $$$$\mathrm{why}....?\:\mathrm{can}\:\mathrm{you}\:\mathrm{explain}\: \\ $$$$\mathrm{why}\:\mathrm{Blue}\:\mathrm{equation}\:\mathrm{is}\:\mathrm{not}\:\mathrm{true}.... \\ $$

Question Number 213776 Answers: 1 Comments: 0

Find the value of the following expression. Ω= (( Im( Li_2 (2)))/(∫_0 ^( (π/2)) ln(sin(x )) dx)) = ?

$$ \\ $$$$\:\:\:\:\:\:\:\:\mathrm{F}{ind}\:\:{the}\:\:{value}\:{of}\:\:{the}\:{following} \\ $$$$\:\:\:\:\:\:\:\:\:\:{expression}. \\ $$$$\:\:\:\:\: \\ $$$$\:\:\:\:\Omega=\:\:\:\frac{\:\mathrm{I}{m}\left(\:\mathrm{Li}_{\mathrm{2}} \:\left(\mathrm{2}\right)\right)}{\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \:\mathrm{ln}\left(\mathrm{sin}\left({x}\:\right)\right)\:{dx}}\:\:=\:? \\ $$

Question Number 213764 Answers: 1 Comments: 0