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AllQuestion and Answers: Page 770

Question Number 139889 Answers: 1 Comments: 1

∫_0 ^4 ((tan^(−1) x)/(x^2 +4x+2))dx=?

$$\int_{\mathrm{0}} ^{\mathrm{4}} \frac{\mathrm{tan}^{−\mathrm{1}} {x}}{{x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{2}}{dx}=? \\ $$

Question Number 139883 Answers: 1 Comments: 0

∫_0 ^π sin^(2n) xdx=?

$$\int_{\mathrm{0}} ^{\pi} \mathrm{sin}\:^{\mathrm{2}{n}} {xdx}=? \\ $$

Question Number 139878 Answers: 2 Comments: 0

I(α) = ∫_α ^α^2 ((sin αx)/x) dx ((dI(α))/dα) =?

$$\:\:\:\:\:\:\:\mathrm{I}\left(\alpha\right)\:=\:\underset{\alpha} {\overset{\alpha^{\mathrm{2}} } {\int}}\:\frac{\mathrm{sin}\:\alpha\mathrm{x}}{\mathrm{x}}\:\mathrm{dx}\: \\ $$$$\:\:\:\:\:\:\:\frac{\mathrm{dI}\left(\alpha\right)}{\mathrm{d}\alpha}\:=? \\ $$

Question Number 139874 Answers: 2 Comments: 0

{ ((xy+24 = (x^3 /y))),((xy−6 = (y^3 /x))) :}

$$\begin{cases}{\mathrm{xy}+\mathrm{24}\:=\:\frac{\mathrm{x}^{\mathrm{3}} }{\mathrm{y}}}\\{\mathrm{xy}−\mathrm{6}\:=\:\frac{\mathrm{y}^{\mathrm{3}} }{\mathrm{x}}}\end{cases} \\ $$

Question Number 139868 Answers: 2 Comments: 0

Question Number 139867 Answers: 1 Comments: 0

a^3 +b^3 +c^3 =0 a^(12) +b^(12) +c^(12) =8 a^6 +b^6 +c^6 =? (ans: 4)

$${a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} =\mathrm{0} \\ $$$${a}^{\mathrm{12}} +{b}^{\mathrm{12}} +{c}^{\mathrm{12}} =\mathrm{8} \\ $$$${a}^{\mathrm{6}} +{b}^{\mathrm{6}} +{c}^{\mathrm{6}} =? \\ $$$$\left({ans}:\:\mathrm{4}\right) \\ $$$$ \\ $$

Question Number 139861 Answers: 2 Comments: 0

((2z+1))^(1/3) + ((8z+4))^(1/5) = (√(16))

$$\sqrt[{\mathrm{3}}]{\mathrm{2}\boldsymbol{{z}}+\mathrm{1}}\:+\:\sqrt[{\mathrm{5}}]{\mathrm{8}\boldsymbol{{z}}+\mathrm{4}}\:=\:\sqrt{\mathrm{16}} \\ $$

Question Number 139851 Answers: 3 Comments: 0

prove ...... Φ= ∫_0 ^( 1) ((ln((1/(1−x))))/( (√x)))dx=4 ln((e/2)) ...✓

$$\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:{prove}\:...... \\ $$$$\:\:\:\:\:\:\Phi=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{ln}\left(\frac{\mathrm{1}}{\mathrm{1}−{x}}\right)}{\:\sqrt{{x}}}{dx}=\mathrm{4}\:{ln}\left(\frac{{e}}{\mathrm{2}}\right)\:...\checkmark \\ $$

Question Number 139842 Answers: 1 Comments: 1

Question Number 139840 Answers: 1 Comments: 1

Question Number 139838 Answers: 1 Comments: 0

If z_1 , z_2 and z_3 are the vertices of a right-angled isos- celes triangle described in counter clock sense and right angled at z_3 , then (z_1 −z_2 )^2 is equal to (A) (z_1 −z_3 )(z_3 −z_2 ) (B) 2(z_1 −z_3 )(z_3 −z_2 ) (C) 3(z_1 −z_3 )(z_3 −z_2 ) (D) 3(z_3 −z_1 )(z_3 −z_2 )

$$\mathrm{If}\:{z}_{\mathrm{1}} ,\:{z}_{\mathrm{2}} \:\mathrm{and}\:{z}_{\mathrm{3}} \:\mathrm{are}\:\mathrm{the}\:\mathrm{vertices}\:\mathrm{of}\:\mathrm{a}\:\mathrm{right}-\mathrm{angled}\:\mathrm{isos}- \\ $$$$\mathrm{celes}\:\mathrm{triangle}\:\mathrm{described}\:\mathrm{in}\:\mathrm{counter}\:\mathrm{clock}\:\mathrm{sense}\:\mathrm{and} \\ $$$$\mathrm{right}\:\mathrm{angled}\:\mathrm{at}\:{z}_{\mathrm{3}} ,\:\mathrm{then}\:\left({z}_{\mathrm{1}} −{z}_{\mathrm{2}} \right)^{\mathrm{2}} \:\mathrm{is}\:\mathrm{equal}\:\mathrm{to}\: \\ $$$$\left(\mathrm{A}\right)\:\left({z}_{\mathrm{1}} −{z}_{\mathrm{3}} \right)\left({z}_{\mathrm{3}} −{z}_{\mathrm{2}} \right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{B}\right)\:\mathrm{2}\left({z}_{\mathrm{1}} −{z}_{\mathrm{3}} \right)\left({z}_{\mathrm{3}} −{z}_{\mathrm{2}} \right) \\ $$$$\left(\mathrm{C}\right)\:\mathrm{3}\left({z}_{\mathrm{1}} −{z}_{\mathrm{3}} \right)\left({z}_{\mathrm{3}} −{z}_{\mathrm{2}} \right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{D}\right)\:\mathrm{3}\left({z}_{\mathrm{3}} −{z}_{\mathrm{1}} \right)\left({z}_{\mathrm{3}} −{z}_{\mathrm{2}} \right) \\ $$

Question Number 139826 Answers: 2 Comments: 0

Σ_(n=0) ^∞ ((sin [(n−1)x])/4^(n+1) )=?

$$\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{sin}\:\left[\left({n}−\mathrm{1}\right){x}\right]}{\mathrm{4}^{{n}+\mathrm{1}} }=? \\ $$

Question Number 139825 Answers: 0 Comments: 1

Question Number 139824 Answers: 1 Comments: 0

Solve in R the following equation: 2∙3^x +5∙4^x =4∙5^x +3∙2^x

$${Solve}\:{in}\:\mathbb{R}\:{the}\:{following}\:{equation}: \\ $$$$\mathrm{2}\centerdot\mathrm{3}^{{x}} +\mathrm{5}\centerdot\mathrm{4}^{{x}} =\mathrm{4}\centerdot\mathrm{5}^{{x}} +\mathrm{3}\centerdot\mathrm{2}^{{x}} \\ $$

Question Number 139818 Answers: 2 Comments: 0

x^4 −22x^2 +x−114=0 x=?

$${x}^{\mathrm{4}} −\mathrm{22}{x}^{\mathrm{2}} +{x}−\mathrm{114}=\mathrm{0} \\ $$$${x}=? \\ $$

Question Number 139815 Answers: 0 Comments: 3

Question Number 139814 Answers: 2 Comments: 0

Question Number 139811 Answers: 2 Comments: 0

𝛗:=∫_0 ^( 1) (√x) ln((1/(1−x)))dx solution: 𝛗:= ∫_0 ^( 1) (√x) Σ_(n=1) ^∞ (x^n /n) dx :=Σ_(n=1) ^∞ (1/n)∫_0 ^( 1) x^(n+(1/2)) dx := Σ_(n=1) ^∞ (1/(n(n+(3/2)))) = (2/3)Σ_(n=1) ^∞ ((1/n)−(1/(n+(3/2)))) :=(2/3){γ −γ+Σ_(n=1) ^∞ ((1/n)−(1/(n+(3/2)))) } := (2/3) γ +(2/3) ψ(1+(3/2))=(2/3) γ +(2/3)((2/3)+ψ((3/2))) :=(2/3) γ +(4/9) +(2/3)(2+ψ((1/2))) :=(2/3) γ +((16)/9) −(2/3) γ−(4/3) ln(2) :=((16)/9) −(4/3) ln(2) .....✓

Question Number 139800 Answers: 2 Comments: 0

(1/3)+(6/(21))+((11)/(147))+((16)/(1029))+((21)/(7203))+((26)/(50421))+…=?

$$\frac{\mathrm{1}}{\mathrm{3}}+\frac{\mathrm{6}}{\mathrm{21}}+\frac{\mathrm{11}}{\mathrm{147}}+\frac{\mathrm{16}}{\mathrm{1029}}+\frac{\mathrm{21}}{\mathrm{7203}}+\frac{\mathrm{26}}{\mathrm{50421}}+\ldots=? \\ $$

Question Number 139799 Answers: 1 Comments: 1

x+(√y)=11 (√x)+y=7 x=? y=?

$${x}+\sqrt{{y}}=\mathrm{11} \\ $$$$\sqrt{{x}}+{y}=\mathrm{7} \\ $$$${x}=?\:\:\:\:\:\:\:{y}=? \\ $$

Question Number 139789 Answers: 2 Comments: 0

......Mathematical ... ... ... Analysis....... evaluation :: F :=∫_0 ^( ∞) e^((−2)/x) sin^2 ((2/x))dx=?

$$\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:......\mathscr{M}{athematical}\:...\:...\:...\:\mathscr{A}{nalysis}....... \\ $$$$\:\:\:\:{evaluation}\:::\:\mathscr{F}\::=\int_{\mathrm{0}} ^{\:\infty} {e}^{\frac{−\mathrm{2}}{{x}}} {sin}^{\mathrm{2}} \left(\frac{\mathrm{2}}{{x}}\right){dx}=? \\ $$$$ \\ $$$$\:\:\:\: \\ $$

Question Number 139776 Answers: 3 Comments: 0

∫ (dx/(2cos(x)+3sin(x)))

$$\int\:\frac{{dx}}{\mathrm{2}{cos}\left({x}\right)+\mathrm{3}{sin}\left({x}\right)} \\ $$