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Question Number 149996 Answers: 3 Comments: 0

(1) ∫ (dx/(1+tanx)) (2)∫ ((√(tanx))/(sinx cosx))dx

$$\left(\mathrm{1}\right)\:\int\:\:\frac{{dx}}{\mathrm{1}+{tanx}} \\ $$$$ \\ $$$$\left(\mathrm{2}\right)\int\:\:\frac{\sqrt{{tanx}}}{{sinx}\:{cosx}}{dx} \\ $$

Question Number 149989 Answers: 1 Comments: 0

By subs u^2 =4+x, evaluate ∫ ((√(4+x))/x) dx

$$\mathrm{By}\:\mathrm{subs}\:{u}^{\mathrm{2}} =\mathrm{4}+{x},\:\mathrm{evaluate}\:\int\:\frac{\sqrt{\mathrm{4}+{x}}}{{x}}\:{dx} \\ $$

Question Number 149986 Answers: 1 Comments: 0

Question Number 149981 Answers: 1 Comments: 0

a full deck of 52 cards contains 13 hearts. Pick 8 cards from the deck at random without replacement. what is the probability that you get no heart?

$$\mathrm{a}\:\mathrm{full}\:\mathrm{deck}\:\mathrm{of}\:\mathrm{52}\:\mathrm{cards}\:\mathrm{contains}\:\mathrm{13} \\ $$$$\:\mathrm{hearts}.\:\mathrm{Pick}\:\mathrm{8}\:\mathrm{cards}\:\mathrm{from}\:\mathrm{the}\:\mathrm{deck} \\ $$$$\mathrm{at}\:\mathrm{random}\:\mathrm{without}\:\mathrm{replacement}. \\ $$$$\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{probability}\:\mathrm{that}\:\mathrm{you}\:\mathrm{get} \\ $$$$\mathrm{no}\:\mathrm{heart}? \\ $$$$ \\ $$

Question Number 149979 Answers: 0 Comments: 2

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Question Number 149962 Answers: 0 Comments: 0

⌊x⌋+⌊y⌋=43.8 and x+y−⌊x⌋=18.4 .Find 100(x+y).

$$\:\lfloor{x}\rfloor+\lfloor{y}\rfloor=\mathrm{43}.\mathrm{8}\:{and}\:{x}+{y}−\lfloor{x}\rfloor=\mathrm{18}.\mathrm{4} \\ $$$$.{Find}\:\mathrm{100}\left({x}+{y}\right). \\ $$

Question Number 149959 Answers: 1 Comments: 3

Question Number 149958 Answers: 4 Comments: 2

lim_(x→0) ((cos(√x)))^(1/x) = ?

$$\underset{\boldsymbol{\mathrm{x}}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\sqrt[{\boldsymbol{\mathrm{x}}}]{\mathrm{cos}\sqrt{\mathrm{x}}}\:=\:? \\ $$

Question Number 149932 Answers: 2 Comments: 0

Question Number 149946 Answers: 6 Comments: 0

Question Number 149944 Answers: 0 Comments: 1

∫_0 ^1 (t^((n−1)/2) /((1+t)^(n+1) ))dt

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{t}^{\frac{{n}−\mathrm{1}}{\mathrm{2}}} }{\left(\mathrm{1}+{t}\right)^{{n}+\mathrm{1}} }{dt} \\ $$

Question Number 149940 Answers: 1 Comments: 1

Question Number 150467 Answers: 2 Comments: 1

Question Number 150462 Answers: 2 Comments: 0

Question Number 149917 Answers: 1 Comments: 0

∫_( 0) ^( 2𝛑) (dt/(4(√2) sint + 6)) = ?

$$\underset{\:\mathrm{0}} {\overset{\:\mathrm{2}\boldsymbol{\pi}} {\int}}\frac{\mathrm{dt}}{\mathrm{4}\sqrt{\mathrm{2}}\:\mathrm{sin}\boldsymbol{\mathrm{t}}\:+\:\mathrm{6}}\:=\:? \\ $$

Question Number 149914 Answers: 0 Comments: 0

show that∫_1 ^2 (((2+6θ^2 −2𝛉^3 )/(𝛉^2 (𝛉^2 +1))))d𝛉=1.606

$$\boldsymbol{{show}}\:\boldsymbol{{that}}\int_{\mathrm{1}} ^{\mathrm{2}} \left(\frac{\mathrm{2}+\mathrm{6}\theta^{\mathrm{2}} −\mathrm{2}\boldsymbol{\theta}^{\mathrm{3}} }{\boldsymbol{\theta}^{\mathrm{2}} \left(\boldsymbol{\theta}^{\mathrm{2}} +\mathrm{1}\right)}\right)\boldsymbol{{d}\theta}=\mathrm{1}.\mathrm{606} \\ $$

Question Number 149903 Answers: 1 Comments: 0

Ω = ∫_0 ^(π/2) ((cos^3 x)/( (√(1−cos^2 x)))) dx

$$\:\Omega\:=\:\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\:\frac{\mathrm{cos}\:^{\mathrm{3}} \mathrm{x}}{\:\sqrt{\mathrm{1}−\mathrm{cos}\:^{\mathrm{2}} \mathrm{x}}}\:\mathrm{dx}\: \\ $$

Question Number 149894 Answers: 1 Comments: 1

Question Number 149891 Answers: 1 Comments: 0

if x;y;z;m;n;p∈R^+ then prove that: Σ_(cyc) ((m(x+y))/( (√((n+2p)x^2 +2nxy+(n+2p)y^2 )))) ≤ ((3m)/( (√(n+p))))

$$\mathrm{if}\:\:\:\mathrm{x};\mathrm{y};\mathrm{z};\mathrm{m};\mathrm{n};\mathrm{p}\in\mathbb{R}^{+} \:\mathrm{then}\:\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\underset{\boldsymbol{\mathrm{cyc}}} {\sum}\:\frac{\mathrm{m}\left(\mathrm{x}+\mathrm{y}\right)}{\:\sqrt{\left(\mathrm{n}+\mathrm{2p}\right)\mathrm{x}^{\mathrm{2}} +\mathrm{2nxy}+\left(\mathrm{n}+\mathrm{2p}\right)\mathrm{y}^{\mathrm{2}} }}\:\leqslant\:\frac{\mathrm{3m}}{\:\sqrt{\mathrm{n}+\mathrm{p}}} \\ $$

Question Number 149889 Answers: 0 Comments: 3

Question Number 149886 Answers: 0 Comments: 0

lim_(x→0^+ ) ((ln^2 (x))/x^2 )(((ln (sin ((x/2))))/(ln (sin (x)))) +((ln 2)/(ln (x)))) =?

$$\:\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\frac{\mathrm{ln}\:^{\mathrm{2}} \left(\mathrm{x}\right)}{\mathrm{x}^{\mathrm{2}} }\left(\frac{\mathrm{ln}\:\left(\mathrm{sin}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\right)}{\mathrm{ln}\:\left(\mathrm{sin}\:\left(\mathrm{x}\right)\right)}\:+\frac{\mathrm{ln}\:\mathrm{2}}{\mathrm{ln}\:\left(\mathrm{x}\right)}\right)\:=? \\ $$

Question Number 149885 Answers: 1 Comments: 0

I_n =∫_0 ^(π/4) (dx/(cos^(2n+1) x)) to show that : ∀ n∈N^∗ , 2nI_n =(2n−1)I_(n−1) +(2^n /( (√2))) (I_n =∫_0 ^(π/4) ((1/(cos^(2n−1) x))×(1/(cos^2 x)))dx)...

$${I}_{{n}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \frac{{dx}}{{cos}^{\mathrm{2}{n}+\mathrm{1}} {x}} \\ $$$${to}\:{show}\:{that}\:: \\ $$$$\forall\:{n}\in\mathbb{N}^{\ast} ,\:\mathrm{2}{nI}_{{n}} =\left(\mathrm{2}{n}−\mathrm{1}\right){I}_{{n}−\mathrm{1}} +\frac{\mathrm{2}^{{n}} }{\:\sqrt{\mathrm{2}}} \\ $$$$\left({I}_{{n}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \left(\frac{\mathrm{1}}{{cos}^{\mathrm{2}{n}−\mathrm{1}} {x}}×\frac{\mathrm{1}}{{cos}^{\mathrm{2}} {x}}\right){dx}\right)... \\ $$

Question Number 149883 Answers: 1 Comments: 0

Prove that ((2+(√5)))^(1/3) +((2−(√5)))^(1/3) is a rational number

$$\mathrm{Prove}\:\mathrm{that}\:\sqrt[{\mathrm{3}}]{\mathrm{2}+\sqrt{\mathrm{5}}}+\sqrt[{\mathrm{3}}]{\mathrm{2}−\sqrt{\mathrm{5}}}\:\mathrm{is} \\ $$$$\mathrm{a}\:\mathrm{rational}\:\mathrm{number} \\ $$

Question Number 149876 Answers: 2 Comments: 0

Question Number 149871 Answers: 1 Comments: 0

lim_(x→2) ((3^(x!) −9)/(x−2))

$${lim}_{{x}\rightarrow\mathrm{2}} \frac{\mathrm{3}^{{x}!} −\mathrm{9}}{{x}−\mathrm{2}} \\ $$

Question Number 149870 Answers: 0 Comments: 3

if x;y;z;m;n∈R^+ then: Σ_(cyc) (b^(−1) /((m(√x) + n(√y))^2 )) ≥ (3/((m + n)^2 ))

$$\mathrm{if}\:\:\:\mathrm{x};\mathrm{y};\mathrm{z};\mathrm{m};\mathrm{n}\in\mathbb{R}^{+} \:\:\mathrm{then}: \\ $$$$\underset{\boldsymbol{\mathrm{cyc}}} {\sum}\:\frac{\mathrm{b}^{−\mathrm{1}} }{\left(\mathrm{m}\sqrt{\mathrm{x}}\:+\:\mathrm{n}\sqrt{\mathrm{y}}\right)^{\mathrm{2}} }\:\geqslant\:\frac{\mathrm{3}}{\left(\mathrm{m}\:+\:\mathrm{n}\right)^{\mathrm{2}} } \\ $$

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