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Question Number 207426 Answers: 1 Comments: 0

Question Number 207424 Answers: 0 Comments: 0

f_n (x):=∫e^((2x)/3) ((cos(x))/( (cos(x)+sin(x))^(n/3) ))dx=...? for n=1, i found f_1 (x)=(3/4)e^((2x)/3) (cos(x)+sin(x))^(2/3) + C is there any ideas for a general case or the case n=2?

$${f}_{{n}} \left({x}\right):=\int{e}^{\frac{\mathrm{2}{x}}{\mathrm{3}}} \frac{{cos}\left({x}\right)}{\:\left({cos}\left({x}\right)+{sin}\left({x}\right)\right)^{\frac{{n}}{\mathrm{3}}} }{dx}=...? \\ $$$${for}\:{n}=\mathrm{1},\:{i}\:{found}\: \\ $$$$\:\:\:\:\:\:{f}_{\mathrm{1}} \left({x}\right)=\frac{\mathrm{3}}{\mathrm{4}}{e}^{\frac{\mathrm{2}{x}}{\mathrm{3}}} \left({cos}\left({x}\right)+{sin}\left({x}\right)\right)^{\frac{\mathrm{2}}{\mathrm{3}}} +\:{C} \\ $$$${is}\:{there}\:{any}\:{ideas}\:{for}\:{a}\:{general}\:{case}\:{or} \\ $$$${the}\:{case}\:{n}=\mathrm{2}? \\ $$

Question Number 207423 Answers: 2 Comments: 0

a_(n+1) = −( / ) + 2a_n a_3 =2. (1) a_5 =−10 (2) a_(2024) > 0 (3) a_(31) −a_(28) (4) a_(101)

$$\:\:\: {a}_{{n}+\mathrm{1}} =\:−\frac{ }{ } +\:\mathrm{2}{a}_{{n}} \: \\ $$$$\:\:\: {a}_{\mathrm{3}} =\mathrm{2}.\: \\ $$$$\:\:\: \\ $$$$ \\ $$$$ \left(\mathrm{1}\right)\:{a}_{\mathrm{5}} =−\mathrm{10} \\ $$$$\:\left(\mathrm{2}\right)\:{a}_{\mathrm{2024}} \:>\:\mathrm{0} \\ $$$$\:\:\left(\mathrm{3}\right)\:{a}_{\mathrm{31}} −{a}_{\mathrm{28}} \: \\ $$$$\:\left(\mathrm{4}\right)\:{a}_{\mathrm{101}} \: \\ $$

Question Number 207416 Answers: 2 Comments: 0

_n _1 + _2 + _3 = ((1/ _1 ) +((1 )/ _2 ) +(1/ _3 )) _2 −7=?

$$\:\: _{\mathrm{n}} \: \\ $$$$ \: \\ $$$$ _{\mathrm{1}} + _{\mathrm{2}} + _{\mathrm{3}} =\: \\ $$$$\: \left(\frac{\mathrm{1}}{ _{\mathrm{1}} }\:+\frac{\mathrm{1} }{ _{\mathrm{2}} }\:+\frac{\mathrm{1}}{ _{\mathrm{3}} }\right)\: _{\mathrm{2}} −\mathrm{7}=? \\ $$

Question Number 207395 Answers: 1 Comments: 0

Geometric series: ((b_4 ∙ b_7 ∙ b_(10) )/(b_1 ∙ b_3 ∙ b_5 )) = 2^(12) find: (b_5 /b_2 ) = ?

$$\mathrm{Geometric}\:\mathrm{series}: \\ $$$$\frac{\mathrm{b}_{\mathrm{4}} \:\centerdot\:\mathrm{b}_{\mathrm{7}} \:\centerdot\:\mathrm{b}_{\mathrm{10}} }{\mathrm{b}_{\mathrm{1}} \:\centerdot\:\mathrm{b}_{\mathrm{3}} \:\centerdot\:\mathrm{b}_{\mathrm{5}} }\:\:=\:\:\mathrm{2}^{\mathrm{12}} \:\:\:\:\:\mathrm{find}:\:\:\:\frac{\mathrm{b}_{\mathrm{5}} }{\mathrm{b}_{\mathrm{2}} }\:\:=\:\:? \\ $$

Question Number 207394 Answers: 1 Comments: 0

(a/b) = (c/d) a^3 − b^3 = 625 c^3 − d^3 = 1 Find: a,b,c,d = ?

$$\frac{\mathrm{a}}{\mathrm{b}}\:\:=\:\:\frac{\mathrm{c}}{\mathrm{d}} \\ $$$$\mathrm{a}^{\mathrm{3}} \:−\:\mathrm{b}^{\mathrm{3}} \:=\:\mathrm{625} \\ $$$$\mathrm{c}^{\mathrm{3}} \:−\:\mathrm{d}^{\mathrm{3}} \:=\:\mathrm{1} \\ $$$$\mathrm{Find}:\:\:\:\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}\:=\:? \\ $$

Question Number 207390 Answers: 0 Comments: 1

Question Number 207389 Answers: 0 Comments: 1

Question Number 207387 Answers: 1 Comments: 0

Let cardE=n , and the set of parts S={(A,B)∈P(E)×P(E) / A∩B=∅} Show that cardS= 3^n

$${Let}\:\:{cardE}={n}\:,\:{and}\:\:{the}\:{set}\:{of}\:{parts} \\ $$$${S}=\left\{\left({A},{B}\right)\in{P}\left({E}\right)×{P}\left({E}\right)\:/\:\:{A}\cap{B}=\varnothing\right\} \\ $$$${Show}\:{that}\:\:{cardS}=\:\mathrm{3}^{{n}} \\ $$

Question Number 207385 Answers: 0 Comments: 1

Find: (√((2,5−(√5))^2 )) − (((1,5−(√5))^3 )^(1/2) ))^(1/3) − (√2) sin ((7π)/4)

$$\mathrm{Find}: \\ $$$$\sqrt{\left(\mathrm{2},\mathrm{5}−\sqrt{\mathrm{5}}\right)^{\mathrm{2}} }\:−\:\sqrt[{\mathrm{3}}]{\left.\left(\mathrm{1},\mathrm{5}−\sqrt{\mathrm{5}}\right)^{\mathrm{3}} \right)^{\frac{\mathrm{1}}{\mathrm{2}}} }\:−\:\sqrt{\mathrm{2}}\:\mathrm{sin}\:\frac{\mathrm{7}\pi}{\mathrm{4}} \\ $$

Question Number 207383 Answers: 0 Comments: 1

∫((ln(x^2 +sin(sin(e^x ))))/( (√(x+tan(ln(x))))))dx

$$\int\frac{{ln}\left({x}^{\mathrm{2}} +{sin}\left({sin}\left({e}^{{x}} \right)\right)\right)}{\:\sqrt{{x}+{tan}\left({ln}\left({x}\right)\right)}}{dx} \\ $$

Question Number 207382 Answers: 1 Comments: 0

Question Number 207381 Answers: 1 Comments: 0

Question Number 207374 Answers: 2 Comments: 0

Show that Σ_(k=0) ^n (C_n ^k )^2 =C_(2n) ^n

$${Show}\:{that}\:\:\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\left({C}_{{n}} ^{{k}} \right)^{\mathrm{2}} ={C}_{\mathrm{2}{n}} ^{{n}} \\ $$

Question Number 207372 Answers: 0 Comments: 6

2 students are passing a test of n questions with the same chance to find each one Show the chance that they both don′t find a same question is ((3/4))^n

$$\mathrm{2}\:{students}\:{are}\:{passing}\: \\ $$$${a}\:{test}\:{of}\:\:{n}\:{questions}\:{with} \\ $$$${the}\:{same}\:{chance}\:{to}\:{find}\:{each}\:{one} \\ $$$${Show}\:\:{the}\:{chance}\:{that}\:{they}\:{both} \\ $$$$\:{don}'{t}\:{find}\:{a}\:{same}\:{question}\:{is}\:\:\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{{n}} \\ $$

Question Number 207402 Answers: 1 Comments: 0

f(x) + 2f((1/x)) = 3x. f ′(x) = ?

$${f}\left({x}\right)\:+\:\mathrm{2}{f}\left(\frac{\mathrm{1}}{{x}}\right)\:=\:\mathrm{3}{x}. \\ $$$$\:{f}\:'\left({x}\right)\:=\:? \\ $$

Question Number 207362 Answers: 2 Comments: 0