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Question Number 216715    Answers: 2   Comments: 0

Find ∫((Sin(((5x )/(2 ))) )/(Sin(((x )/(2 ))) )) .dx

$$\:\:\:\:\boldsymbol{{F}}{ind}\:\int\frac{\boldsymbol{{S}}{in}\left(\frac{\mathrm{5}{x}\:}{\mathrm{2}\:}\right)\:\:}{\boldsymbol{{S}}{in}\left(\frac{{x}\:}{\mathrm{2}\:}\right)\:\:\:\:\:}\:.\boldsymbol{{dx}}\:\:\: \\ $$

Question Number 216710    Answers: 0   Comments: 0

let y_1 , y_2 , y_(3 ) ... y_p be fixed positive number consider the sequences s_n = ((y_1 ^n +y_2 ^n +y_3 ^n +...+y_p ^n )/p) and x_(n ) = s_n ^(1/n) , n∈N show that {x_n } is monotonically increasing

$$\:\:\:\mathrm{let}\:{y}_{\mathrm{1}} \:,\:{y}_{\mathrm{2}} \:,\:{y}_{\mathrm{3}\:\:} ...\:{y}_{{p}} \:\mathrm{be}\:\mathrm{fixed}\:\mathrm{positive}\:\mathrm{number} \\ $$$$\:\:\:\mathrm{consider}\:\mathrm{the}\:\mathrm{sequences} \\ $$$$\:{s}_{{n}} \:=\:\:\frac{{y}_{\mathrm{1}} ^{{n}} +{y}_{\mathrm{2}} ^{{n}} +{y}_{\mathrm{3}} ^{{n}} +...+{y}_{{p}} ^{{n}} }{{p}}\:\:\:{and}\:{x}_{{n}\:} =\:{s}_{{n}} ^{\mathrm{1}/{n}} \:\:,\:\:\:{n}\in\mathbb{N} \\ $$$$\:\:\:\mathrm{show}\:\mathrm{that}\:\left\{{x}_{{n}} \right\}\:\mathrm{is}\:\mathrm{monotonically}\:\mathrm{increasing} \\ $$

Question Number 216706    Answers: 1   Comments: 4

∫ (dz/(1+sin(z)cos(z)))= ∫ ((sec^2 (z)dz)/(sec^2 (z)+tan(z))) multiply sec^2 (z) sec^2 (z)=1+tan^2 (z) ∫ ((sec^2 (z) dz)/(1+tan(z)+tan^2 (z))) s=tan(z) ds=sec^2 (z)dz ∫ (ds/(s^2 +s+1))=∫ (ds/((s+(1/2))^2 +(3/4))) = ∫ (dq/(q^2 +(3/4))) q=s+(1/2) (2/( (√3)))∫ (1/(w^2 +1)) dw w=((2q)/( (√3))) (2/( (√3)))tan^(−1) (w)+C → (2/( (√3)))tan^(−1) (((2q)/( (√3))))+C q=s+(1/4) (2/( (√3)))tan^(−1) ((2/( (√3)))(s+(1/2)))+C s=tan(z) ∴ ∫ (dz/(1+sin(z)cos(z)))=(2/( (√3)))tan^(−1) (((2tan(z)+1)/( (√3))))+C ∫_0 ^( 2π) =((4π)/( (√3)))

$$\int\:\:\:\frac{\mathrm{d}{z}}{\mathrm{1}+\mathrm{sin}\left({z}\right)\mathrm{cos}\left({z}\right)}= \\ $$$$\int\:\:\:\frac{\mathrm{sec}^{\mathrm{2}} \left({z}\right)\mathrm{d}{z}}{\mathrm{sec}^{\mathrm{2}} \left({z}\right)+\mathrm{tan}\left({z}\right)}\:\mathrm{multiply}\:\mathrm{sec}^{\mathrm{2}} \left({z}\right) \\ $$$$\mathrm{sec}^{\mathrm{2}} \left({z}\right)=\mathrm{1}+\mathrm{tan}^{\mathrm{2}} \left({z}\right) \\ $$$$\int\:\:\:\frac{\mathrm{sec}^{\mathrm{2}} \left({z}\right)\:\mathrm{d}{z}}{\mathrm{1}+\mathrm{tan}\left({z}\right)+\mathrm{tan}^{\mathrm{2}} \left({z}\right)} \\ $$$${s}=\mathrm{tan}\left({z}\right) \\ $$$$\mathrm{d}{s}=\mathrm{sec}^{\mathrm{2}} \left({z}\right)\mathrm{d}{z} \\ $$$$\int\:\:\frac{\mathrm{d}{s}}{{s}^{\mathrm{2}} +{s}+\mathrm{1}}=\int\:\:\frac{\mathrm{d}{s}}{\left({s}+\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} +\frac{\mathrm{3}}{\mathrm{4}}}\:=\:\int\:\:\:\frac{\mathrm{d}{q}}{{q}^{\mathrm{2}} +\frac{\mathrm{3}}{\mathrm{4}}} \\ $$$${q}={s}+\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\frac{\mathrm{2}}{\:\sqrt{\mathrm{3}}}\int\:\frac{\mathrm{1}}{{w}^{\mathrm{2}} +\mathrm{1}}\:\mathrm{d}{w} \\ $$$${w}=\frac{\mathrm{2}{q}}{\:\sqrt{\mathrm{3}}} \\ $$$$\frac{\mathrm{2}}{\:\sqrt{\mathrm{3}}}\mathrm{tan}^{−\mathrm{1}} \left({w}\right)+{C}\:\rightarrow\:\frac{\mathrm{2}}{\:\sqrt{\mathrm{3}}}\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{2}{q}}{\:\sqrt{\mathrm{3}}}\right)+{C} \\ $$$${q}={s}+\frac{\mathrm{1}}{\mathrm{4}} \\ $$$$\frac{\mathrm{2}}{\:\sqrt{\mathrm{3}}}\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{2}}{\:\sqrt{\mathrm{3}}}\left({s}+\frac{\mathrm{1}}{\mathrm{2}}\right)\right)+{C} \\ $$$${s}=\mathrm{tan}\left({z}\right) \\ $$$$\therefore\:\int\:\:\frac{\mathrm{d}{z}}{\mathrm{1}+\mathrm{sin}\left({z}\right)\mathrm{cos}\left({z}\right)}=\frac{\mathrm{2}}{\:\sqrt{\mathrm{3}}}\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{2tan}\left({z}\right)+\mathrm{1}}{\:\sqrt{\mathrm{3}}}\right)+{C} \\ $$$$\int_{\mathrm{0}} ^{\:\mathrm{2}\pi} \:=\frac{\mathrm{4}\pi}{\:\sqrt{\mathrm{3}}} \\ $$

Question Number 216703    Answers: 1   Comments: 0

Question Number 216698    Answers: 2   Comments: 0

lim_(x→+∞) ^3 (√(x+1)) −^3 (√x) =^? 0

$$\underset{{x}\rightarrow+\infty} {\mathrm{lim}}\:^{\mathrm{3}} \sqrt{{x}+\mathrm{1}}\:−^{\mathrm{3}} \sqrt{{x}}\:\overset{?} {=}\:\mathrm{0} \\ $$

Question Number 216695    Answers: 1   Comments: 0

∫_0 ^(2π) (dx/(1+sinxcosx))=^? ((4πln2)/( (√3)))

$$\int_{\mathrm{0}} ^{\mathrm{2}\pi} \frac{{dx}}{\mathrm{1}+{sinxcosx}}\overset{?} {=}\:\frac{\mathrm{4}\pi{ln}\mathrm{2}}{\:\sqrt{\mathrm{3}}}\:\: \\ $$

Question Number 216694    Answers: 4   Comments: 0

Prove that ^3 (√((√5)+2)) −^3 (√((√5)−2)) =1

$${Prove}\:{that}\:\:^{\mathrm{3}} \sqrt{\sqrt{\mathrm{5}}+\mathrm{2}}\:−^{\mathrm{3}} \sqrt{\sqrt{\mathrm{5}}−\mathrm{2}}\:=\mathrm{1} \\ $$

Question Number 216679    Answers: 0   Comments: 0

Question Number 216672    Answers: 0   Comments: 0

Question Number 216670    Answers: 3   Comments: 0

Question Number 216664    Answers: 1   Comments: 0

Solve for non-negative integers: n^3 =3m(m+2n+1)

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{non}-\mathrm{negative}\:\mathrm{integers}: \\ $$$$\:\:\:\mathrm{n}^{\mathrm{3}} =\mathrm{3m}\left(\mathrm{m}+\mathrm{2n}+\mathrm{1}\right) \\ $$

Question Number 216659    Answers: 1   Comments: 0

Question Number 216656    Answers: 0   Comments: 0

Question Number 216655    Answers: 1   Comments: 0

Question Number 216647    Answers: 0   Comments: 0

∫_0 ^(π/2) (dx/((acos^2 x+bsin^2 x)^n ))

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{{dx}}{\left({acos}^{\mathrm{2}} {x}+{bsin}^{\mathrm{2}} {x}\right)^{{n}} } \\ $$

Question Number 216646    Answers: 3   Comments: 0

Let a,b,c be positive roots of an cubic equation such that ab+bc+ac+abc=4, show using vieta′s relations that (a/(a+2))+(b/(b+2))+(c/(c+2))=1

$$\mathrm{Let}\:{a},{b},{c}\:\mathrm{be}\:\mathrm{positive}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{an}\:\mathrm{cubic}\:\mathrm{equation}\:\mathrm{such}\:\mathrm{that} \\ $$$${ab}+{bc}+{ac}+{abc}=\mathrm{4},\:\mathrm{show}\:\mathrm{using}\:\mathrm{vieta}'\mathrm{s}\:\mathrm{relations}\:\mathrm{that} \\ $$$$\frac{{a}}{{a}+\mathrm{2}}+\frac{{b}}{{b}+\mathrm{2}}+\frac{{c}}{{c}+\mathrm{2}}=\mathrm{1} \\ $$

Question Number 216645    Answers: 0   Comments: 0

Question Number 216665    Answers: 1   Comments: 1

Question Number 216638    Answers: 1   Comments: 1

without using LHopital rule evalute lim_(x→0) ((ln(1−x)−sin(x) )/(1−cox^2 (x)))

$$\:\:\boldsymbol{{without}}\:\boldsymbol{{using}}\:\boldsymbol{{LHopital}} \\ $$$$\:\:\:\boldsymbol{{rule}}\:\boldsymbol{{evalute}}\: \\ $$$$\:\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\boldsymbol{{ln}}\left(\mathrm{1}−{x}\right)−\boldsymbol{{sin}}\left(\boldsymbol{{x}}\right)\:}{\mathrm{1}−\boldsymbol{{cox}}^{\mathrm{2}} \left(\boldsymbol{{x}}\right)} \\ $$

Question Number 216630    Answers: 2   Comments: 0

the circles x² + y² -4x -2y +3 =0 and x² +y² + 2x +4y -3 =0 touches each other Find the coordinates of the point of contact

the circles x² + y² -4x -2y +3 =0 and x² +y² + 2x +4y -3 =0 touches each other Find the coordinates of the point of contact

Question Number 216621    Answers: 2   Comments: 0

Question Number 216618    Answers: 2   Comments: 0

∫_( 0) ^( 2π) (√(1 − cos^2 x)) dx Is the answer 0 or 4????

$$\int_{\:\mathrm{0}} ^{\:\mathrm{2}\pi} \:\sqrt{\mathrm{1}\:\:−\:\:\mathrm{cos}^{\mathrm{2}} \mathrm{x}}\:\:\mathrm{dx} \\ $$$$\mathrm{Is}\:\mathrm{the}\:\mathrm{answer}\:\:\mathrm{0}\:\:\:\mathrm{or}\:\:\:\:\mathrm{4}???? \\ $$

Question Number 216615    Answers: 2   Comments: 0

∫_0 ^1 sin nπx J_0 (j_(0m) x)dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{sin}\:{n}\pi{x}\:{J}_{\mathrm{0}} \left({j}_{\mathrm{0}{m}} {x}\right){dx} \\ $$

Question Number 216613    Answers: 0   Comments: 1

∫_a ^( x) ((1−bln (x/a))/( (√(1−(1−bln (x/a))^2 )))) dx

$$\int_{{a}} ^{\:{x}} \frac{\mathrm{1}−{b}\mathrm{ln}\:\frac{{x}}{{a}}}{\:\sqrt{\mathrm{1}−\left(\mathrm{1}−{b}\mathrm{ln}\:\frac{{x}}{{a}}\right)^{\mathrm{2}} }}\:{dx} \\ $$

Question Number 216607    Answers: 1   Comments: 0

Question Number 216596    Answers: 1   Comments: 0

∫∫_(D) ((sin(x^2 +y^2 )+tan(x^2 +y^2 ))/(cos(x^2 +y^2 )+tan(x^2 +y^2 )))dxdy D=[0,(π/4)]×[0,(π/4)]

$$\underset{\mathcal{D}} {\int\int}\:\:\:\frac{\mathrm{sin}\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)+\mathrm{tan}\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)}{\mathrm{cos}\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)+\mathrm{tan}\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)}\mathrm{d}{x}\mathrm{d}{y} \\ $$$$\mathcal{D}=\left[\mathrm{0},\frac{\pi}{\mathrm{4}}\right]×\left[\mathrm{0},\frac{\pi}{\mathrm{4}}\right] \\ $$

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