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Question Number 66336 Answers: 0 Comments: 0

calculate ∫_0 ^(π/4) ((tanx)/((√2)cosx +2sin^2 x))dx

$${calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\frac{{tanx}}{\sqrt{\mathrm{2}}{cosx}\:+\mathrm{2}{sin}^{\mathrm{2}} {x}}{dx} \\ $$

Question Number 66335 Answers: 0 Comments: 1

find ∫_(−(π/6)) ^(π/6) ((1+tanx)/(1+sin(2x)))dx

$${find}\:\:\int_{−\frac{\pi}{\mathrm{6}}} ^{\frac{\pi}{\mathrm{6}}} \:\frac{\mathrm{1}+{tanx}}{\mathrm{1}+{sin}\left(\mathrm{2}{x}\right)}{dx} \\ $$

Question Number 66334 Answers: 0 Comments: 2

let I =∫_0 ^(π/4) e^(−2t) cos^4 t dt and J=∫_0 ^(π/4) e^(−2t) sin^4 tdt 1)calculate I+J and I−J 2) find the value of I and J.

$${let}\:{I}\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:{e}^{−\mathrm{2}{t}} \:{cos}^{\mathrm{4}} {t}\:{dt}\:{and}\:{J}=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:{e}^{−\mathrm{2}{t}} \:{sin}^{\mathrm{4}} {tdt} \\ $$$$\left.\mathrm{1}\right){calculate}\:\:{I}+{J}\:{and}\:{I}−{J} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:{I}\:{and}\:{J}. \\ $$

Question Number 66332 Answers: 0 Comments: 0

let A_n =∫_0 ^1 x^n (√(1−x))dx 1)calculate A_0 and A_1 2)prove that ∀n∈N^★ (3+2n)A_n =2nA_(n−1) 3) find A_n interms of n.

$${let}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \:{x}^{{n}} \sqrt{\mathrm{1}−{x}}{dx} \\ $$$$\left.\mathrm{1}\right){calculate}\:{A}_{\mathrm{0}} \:{and}\:{A}_{\mathrm{1}} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:\forall{n}\in{N}^{\bigstar} \:\:\:\:\left(\mathrm{3}+\mathrm{2}{n}\right){A}_{{n}} =\mathrm{2}{nA}_{{n}−\mathrm{1}} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{A}_{{n}} \:{interms}\:{of}\:{n}. \\ $$

Question Number 66330 Answers: 0 Comments: 1

let I_n =∫_0 ^1 x^n e^(−x) dx with n integr natural 1) calculate I_0 , I_1 and I_2 2)find arelation between I_n and I_n 3) find I_n interms of n.

$${let}\:{I}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \:{x}^{{n}} \:{e}^{−{x}} \:{dx}\:\:\:\:{with}\:{n}\:{integr}\:{natural} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{I}_{\mathrm{0}} \:,\:{I}_{\mathrm{1}} \:{and}\:{I}_{\mathrm{2}} \\ $$$$\left.\mathrm{2}\right){find}\:{arelation}\:{between}\:{I}_{{n}} \:{and}\:{I}_{{n}} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{I}_{{n}} \:{interms}\:{of}\:{n}. \\ $$

Question Number 66328 Answers: 0 Comments: 1

calculate ∫_0 ^1 (dt/((1+t^2 )^3 ))

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{dt}}{\left(\mathrm{1}+{t}^{\mathrm{2}} \right)^{\mathrm{3}} } \\ $$

Question Number 66326 Answers: 0 Comments: 4

calculate ∫_0 ^1 ((x^4 +1)/(x^6 +1))dx

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{x}^{\mathrm{4}} \:+\mathrm{1}}{{x}^{\mathrm{6}} +\mathrm{1}}{dx} \\ $$

Question Number 66324 Answers: 0 Comments: 1

find nature of the serie Σ_(n=1) ^∞ (((−1)^(n+1) )/(2^n +ln(n)))

$${find}\:{nature}\:{of}\:{the}\:{serie}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}+\mathrm{1}} }{\mathrm{2}^{{n}} \:+{ln}\left({n}\right)} \\ $$

Question Number 66323 Answers: 0 Comments: 0

find lim_(x→0^+ ) ((x(1+cosx)−2tanx)/(2x−sinx−tanx))

$${find}\:{lim}_{{x}\rightarrow\mathrm{0}^{+} } \:\:\:\:\frac{{x}\left(\mathrm{1}+{cosx}\right)−\mathrm{2}{tanx}}{\mathrm{2}{x}−{sinx}−{tanx}} \\ $$

Question Number 66322 Answers: 0 Comments: 0

let f(x)=(cosx)^(1/x) ( 1) prove that f(x)∼1−(x/2)+(x^2 /8) ( x→0) (2)ptove that f^′ (x)∼−(2/π) e^(((1/x)−1)ln(cosx)) (x→(π/2))

$${let}\:{f}\left({x}\right)=\left({cosx}\right)^{\frac{\mathrm{1}}{{x}}} \:\left(\:\mathrm{1}\right)\:\:{prove}\:{that}\:{f}\left({x}\right)\sim\mathrm{1}−\frac{{x}}{\mathrm{2}}+\frac{{x}^{\mathrm{2}} }{\mathrm{8}}\:\:\left(\:{x}\rightarrow\mathrm{0}\right) \\ $$$$\left(\mathrm{2}\right){ptove}\:{that}\:{f}^{'} \left({x}\right)\sim−\frac{\mathrm{2}}{\pi}\:{e}^{\left(\frac{\mathrm{1}}{{x}}−\mathrm{1}\right){ln}\left({cosx}\right)} \:\:\left({x}\rightarrow\frac{\pi}{\mathrm{2}}\right) \\ $$

Question Number 66321 Answers: 0 Comments: 2

find lim_(x→0^+ ) (tan((π/(2+x))))^x

$${find}\:{lim}_{{x}\rightarrow\mathrm{0}^{+} } \:\:\:\left({tan}\left(\frac{\pi}{\mathrm{2}+{x}}\right)\right)^{{x}} \\ $$

Question Number 66319 Answers: 0 Comments: 0

find lim_(x→+∞) (((a^(1/x) +2b^(1/x) +3c^(1/x) )/6))^x

$${find}\:{lim}_{{x}\rightarrow+\infty} \:\:\:\:\left(\frac{{a}^{\frac{\mathrm{1}}{{x}}} \:+\mathrm{2}{b}^{\frac{\mathrm{1}}{{x}}} +\mathrm{3}{c}^{\frac{\mathrm{1}}{{x}}} }{\mathrm{6}}\right)^{{x}} \\ $$

Question Number 66318 Answers: 0 Comments: 4

find lim_(x→0) (((1+x)/(1−x)))^(1/(sinx))

$${find}\:{lim}_{{x}\rightarrow\mathrm{0}} \left(\frac{\mathrm{1}+{x}}{\mathrm{1}−{x}}\right)^{\frac{\mathrm{1}}{{sinx}}} \\ $$

Question Number 66317 Answers: 0 Comments: 2

calculate lim_(x→0) ((ln(cosx))/(1−cos(2x)))

$${calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\frac{{ln}\left({cosx}\right)}{\mathrm{1}−{cos}\left(\mathrm{2}{x}\right)} \\ $$

Question Number 66316 Answers: 1 Comments: 1

lim_(x→(π/2)) ((ln(sin^2 x))/(((π/2)−x)^2 ))

$${lim}_{{x}\rightarrow\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{ln}\left({sin}^{\mathrm{2}} {x}\right)}{\left(\frac{\pi}{\mathrm{2}}−{x}\right)^{\mathrm{2}} } \\ $$$$ \\ $$

Question Number 66325 Answers: 0 Comments: 3

calculate ∫_0 ^(π/4) cos^4 x sin^2 x dx

$${calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:{cos}^{\mathrm{4}} {x}\:{sin}^{\mathrm{2}} {x}\:{dx} \\ $$

Question Number 66308 Answers: 0 Comments: 1

find ∫ (dx/((x+3)(√(−x^2 −4x))))

$${find}\:\int\:\:\:\frac{{dx}}{\left({x}+\mathrm{3}\right)\sqrt{−{x}^{\mathrm{2}} −\mathrm{4}{x}}} \\ $$

Question Number 66310 Answers: 1 Comments: 1

Question Number 66309 Answers: 0 Comments: 2

calculate ∫ (dx/((x^2 −1)(√(x^2 +2))))

$${calculate}\:\:\int\:\:\:\:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} −\mathrm{1}\right)\sqrt{{x}^{\mathrm{2}} +\mathrm{2}}} \\ $$

Question Number 66304 Answers: 1 Comments: 0

calculate lim_(x→0) ((ln(x+1+sin(πx)))/(xsin(2x)))

$${calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\frac{{ln}\left({x}+\mathrm{1}+{sin}\left(\pi{x}\right)\right)}{{xsin}\left(\mathrm{2}{x}\right)} \\ $$

Question Number 66302 Answers: 0 Comments: 3

solved the general quintic, despite whatever proof that it cant be solved in a simple way!

$${solved}\:{the}\:{general}\:{quintic}, \\ $$$${despite}\:{whatever}\:{proof}\:{that}\:{it} \\ $$$${cant}\:{be}\:{solved}\:{in}\:{a}\:{simple}\:{way}! \\ $$

Question Number 66293 Answers: 0 Comments: 3

What do we mean by ∫_(−∞) ^(+∞) f(x) dx?

$${What}\:{do}\:{we}\:{mean}\:{by}\:\: \\ $$$$\:\:\int_{−\infty} ^{+\infty} {f}\left({x}\right)\:{dx}? \\ $$

Question Number 66290 Answers: 1 Comments: 0

(x^2 )^(1/(√3)) + x^(√3) − 392 = 0

$$\:\sqrt[{\sqrt{\mathrm{3}}}]{\boldsymbol{\mathrm{x}}^{\mathrm{2}} }\:+\:\boldsymbol{\mathrm{x}}^{\sqrt{\mathrm{3}}} \:−\:\mathrm{392}\:=\:\mathrm{0} \\ $$

Question Number 66287 Answers: 1 Comments: 0

10^(10^(10^(.∙^(.10) ) ) ) =?

$$\mathrm{10}^{\mathrm{10}^{\mathrm{10}^{.\centerdot^{.\mathrm{10}} } } } =?\: \\ $$

Question Number 66285 Answers: 2 Comments: 0

What is the difference between lim_(x→2^− ) and lim_(x→2^+ )

$${What}\:{is}\:{the}\:{difference}\:{between} \\ $$$$\:\:\underset{{x}\rightarrow\mathrm{2}^{−} } {{lim}}\:\:{and} \\ $$$$\underset{{x}\rightarrow\mathrm{2}^{+} } {{lim}} \\ $$

Question Number 66279 Answers: 1 Comments: 1

Value of maximum f(x)=^2 log(x+5)+^2 log(3−x) is... a.4 b.8 c. 12 d. 15 e. 16

$$\mathrm{Value}\:\mathrm{of}\:\mathrm{maximum} \\ $$$${f}\left({x}\right)=^{\mathrm{2}} \mathrm{log}\left({x}+\mathrm{5}\right)+^{\mathrm{2}} \mathrm{log}\left(\mathrm{3}−{x}\right) \\ $$$$\mathrm{is}... \\ $$$$\mathrm{a}.\mathrm{4} \\ $$$$\mathrm{b}.\mathrm{8} \\ $$$$\mathrm{c}.\:\mathrm{12} \\ $$$$\mathrm{d}.\:\mathrm{15} \\ $$$$\mathrm{e}.\:\mathrm{16} \\ $$

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