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Question Number 208241 Answers: 1 Comments: 0

Solve for p, q, r p+q+r=α p^2 +q^2 +r^2 =β pq=r

$$\mathrm{Solve}\:\mathrm{for}\:{p},\:{q},\:{r} \\ $$$${p}+{q}+{r}=\alpha \\ $$$${p}^{\mathrm{2}} +{q}^{\mathrm{2}} +{r}^{\mathrm{2}} =\beta \\ $$$${pq}={r} \\ $$

Question Number 208238 Answers: 1 Comments: 0

Show that (π/4) < ∫_0 ^1 (√(1−x^4 ))dx using x = sint show that ∫_0 ^1 (√(1−x^4 ))dx<((2(√2))/3) using (∫_0 ^1 f(x)g(x)dx)^2 <∫_0 ^1 (f(x))^2 dx∫_0 ^1 (g(x))^2 dx

$$\mathrm{S}{how}\:{that} \\ $$$$\frac{\pi}{\mathrm{4}}\:<\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\sqrt{\mathrm{1}−{x}^{\mathrm{4}} }{dx}\:{using}\:{x}\:=\:{sint} \\ $$$${show}\:{that}\:\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt{\mathrm{1}−{x}^{\mathrm{4}} }{dx}<\frac{\mathrm{2}\sqrt{\mathrm{2}}}{\mathrm{3}} \\ $$$${using}\:\left(\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){g}\left({x}\right){dx}\right)^{\mathrm{2}} <\int_{\mathrm{0}} ^{\mathrm{1}} \left({f}\left({x}\right)\right)^{\mathrm{2}} {dx}\int_{\mathrm{0}} ^{\mathrm{1}} \left({g}\left({x}\right)\right)^{\mathrm{2}} {dx} \\ $$

Question Number 208235 Answers: 2 Comments: 0

Question Number 208218 Answers: 1 Comments: 4

a_n numbers series If S_(16) − S_(13) = S_(106) − S_(103) Find: ((3a_3 + 4a_4 + 5a_5 )/(2a_(12) )) = ?

$$\mathrm{a}_{\boldsymbol{\mathrm{n}}} \:\:\mathrm{numbers}\:\mathrm{series} \\ $$$$\mathrm{If}\:\:\mathrm{S}_{\mathrm{16}} \:−\:\mathrm{S}_{\mathrm{13}} \:\:=\:\:\mathrm{S}_{\mathrm{106}} \:−\:\mathrm{S}_{\mathrm{103}} \\ $$$$\mathrm{Find}:\:\:\:\:\frac{\mathrm{3a}_{\mathrm{3}} \:+\:\mathrm{4a}_{\mathrm{4}} \:+\:\mathrm{5a}_{\mathrm{5}} }{\mathrm{2a}_{\mathrm{12}} }\:\:=\:\:? \\ $$

Question Number 208217 Answers: 3 Comments: 0

1^2 +2^2 +3^2 +5^2 +8^2 +13^2 +21^2 =?

$$\mathrm{1}^{\mathrm{2}} +\mathrm{2}^{\mathrm{2}} +\mathrm{3}^{\mathrm{2}} +\mathrm{5}^{\mathrm{2}} +\mathrm{8}^{\mathrm{2}} +\mathrm{13}^{\mathrm{2}} +\mathrm{21}^{\mathrm{2}} =? \\ $$

Question Number 208215 Answers: 2 Comments: 0

If (1/R) = (1/R_1 ) + (1/R_2 ) [R_1 , R_2 > 0] and R_1 + R_2 = C (Constant) then prove that R will be maximum when R_1 = R_2 .

Question Number 208205 Answers: 1 Comments: 0

∫ (x^3 . 5^(2x^2 −2) ) dx =?

$$\:\:\:\int\:\left({x}^{\mathrm{3}} .\:\mathrm{5}^{\mathrm{2}{x}^{\mathrm{2}} −\mathrm{2}} \:\right)\:{dx}\:=? \\ $$

Question Number 208199 Answers: 2 Comments: 0

Question Number 208194 Answers: 2 Comments: 0

I_n = ∫_(0 ) ^∞ (1/((1+x^2 )^n ))dx prove that Σ_(n=1) ^∞ (I_n /n) = π

$$\:\:\:\:\:\:\:\:{I}_{{n}} \:=\:\:\int_{\mathrm{0}\:} ^{\infty} \frac{\mathrm{1}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{{n}} }{dx} \\ $$$$\:\:{prove}\:{that}\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{I}_{{n}} }{{n}}\:\:=\:\:\pi \\ $$

Question Number 208187 Answers: 1 Comments: 0

Find: 1,03^(200) = ?

$$\mathrm{Find}:\:\:\:\mathrm{1},\mathrm{03}^{\mathrm{200}} \:=\:? \\ $$

Question Number 208178 Answers: 5 Comments: 3

Question Number 208176 Answers: 1 Comments: 1

Find the value of the folloing integral. determinant ((( 𝛀=∫_0 ^( (𝛑/2)) (( 1)/(1 + (( cosx))^(1/3) )) dx = ? )))

$$ \\ $$$$\:\:\:\:\:\boldsymbol{{Find}}\:\boldsymbol{{the}}\:\boldsymbol{{value}}\:\boldsymbol{{of}}\:\boldsymbol{{the}} \\ $$$$\:\:\:\:\:\:\:\boldsymbol{{folloing}}\:\boldsymbol{{integral}}. \\ $$$$\:\:\:\:\:\:\: \\ $$$$\begin{array}{|c|}{\:\:\:\boldsymbol{\Omega}=\int_{\mathrm{0}} ^{\:\frac{\boldsymbol{\pi}}{\mathrm{2}}} \:\frac{\:\mathrm{1}}{\mathrm{1}\:+\:\sqrt[{\mathrm{3}}]{\:\boldsymbol{{cosx}}}}\:\boldsymbol{{dx}}\:=\:?\:\:}\\\hline\end{array} \\ $$$$\:\:\:\:\:\:\: \\ $$

Question Number 208173 Answers: 1 Comments: 1

(5/(−∞))=?

$$\frac{\mathrm{5}}{−\infty}=? \\ $$

Question Number 208167 Answers: 2 Comments: 0

y = 3 cos^2 α + 2 cos α find: max(y) = ?

$$\mathrm{y}\:=\:\mathrm{3}\:\mathrm{cos}^{\mathrm{2}} \:\alpha\:+\:\mathrm{2}\:\mathrm{cos}\:\alpha \\ $$$$\mathrm{find}:\:\:\:\mathrm{max}\left(\mathrm{y}\right)\:=\:? \\ $$

Question Number 208164 Answers: 2 Comments: 0

Solve for x: x^2 +x^2 (1−x^2 )+x^2 (1−x^2 )^2 +x^2 (1−x^2 )^3 +...+x^2 (1−x^2 )^(100) =1

$${Solve}\:{for}\:{x}: \\ $$$${x}^{\mathrm{2}} +{x}^{\mathrm{2}} \left(\mathrm{1}−{x}^{\mathrm{2}} \right)+{x}^{\mathrm{2}} \left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{\mathrm{2}} +{x}^{\mathrm{2}} \left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{\mathrm{3}} +...+{x}^{\mathrm{2}} \left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{\mathrm{100}} =\mathrm{1} \\ $$

Question Number 208149 Answers: 3 Comments: 0

Question Number 208148 Answers: 2 Comments: 0

Question Number 208139 Answers: 1 Comments: 0

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Question Number 208135 Answers: 1 Comments: 0

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$$\:\:\:\downharpoonleft\underline{\:} \\ $$

Question Number 208134 Answers: 0 Comments: 0

Question Number 208140 Answers: 1 Comments: 0

∫_0 ^π (dx/(1+((sin x))^(1/3) ))=? exact result required

$$\underset{\mathrm{0}} {\overset{\pi} {\int}}\frac{{dx}}{\mathrm{1}+\sqrt[{\mathrm{3}}]{\mathrm{sin}\:{x}}}=? \\ $$$$\mathrm{exact}\:\mathrm{result}\:\mathrm{required} \\ $$

Question Number 208130 Answers: 1 Comments: 0

(1/(cos x−cos 3x)) + (1/(cos x−cos 5x)) + (1/(cos x−cos 7x)) + (1/(cos x−cos 11x))=?

$$\:\:\frac{\mathrm{1}}{\mathrm{cos}\:\mathrm{x}−\mathrm{cos}\:\mathrm{3x}}\:+\:\frac{\mathrm{1}}{\mathrm{cos}\:\mathrm{x}−\mathrm{cos}\:\mathrm{5x}}\:+ \\ $$$$\:\frac{\mathrm{1}}{\mathrm{cos}\:\mathrm{x}−\mathrm{cos}\:\mathrm{7x}}\:+\:\frac{\mathrm{1}}{\mathrm{cos}\:\mathrm{x}−\mathrm{cos}\:\mathrm{11x}}=?\: \\ $$

Question Number 208129 Answers: 2 Comments: 0

$$\:\:\:\:\:\underbrace{\pm \cancel{} } \\ $$

Question Number 208128 Answers: 0 Comments: 0

Question Number 208122 Answers: 0 Comments: 0

Question Number 208121 Answers: 0 Comments: 2

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