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Question Number 151612 Answers: 0 Comments: 0

∫_0 ^e (x/( (√(x−ln(x)))))dx

$$\int_{\mathrm{0}} ^{{e}} \frac{{x}}{\:\sqrt{{x}−{ln}\left({x}\right)}}{dx} \\ $$

Question Number 151609 Answers: 1 Comments: 0

Question Number 151602 Answers: 6 Comments: 0

Question Number 151599 Answers: 1 Comments: 0

Ω =∫_( 0) ^( ∞) cos(x^n ) dx = ?

$$\Omega\:=\underset{\:\mathrm{0}} {\overset{\:\infty} {\int}}\mathrm{cos}\left(\mathrm{x}^{\boldsymbol{\mathrm{n}}} \right)\:\mathrm{dx}\:=\:? \\ $$

Question Number 151596 Answers: 1 Comments: 0

Question Number 151587 Answers: 1 Comments: 0

∫sin^(−1) (√(x/(a+x))) dx

$$\int\mathrm{sin}^{−\mathrm{1}} \sqrt{\frac{\mathrm{x}}{\mathrm{a}+\mathrm{x}}}\:\mathrm{dx} \\ $$

Question Number 151586 Answers: 0 Comments: 0

∫e^(tan^(−1) x) (((1+x+x^2 )/(x^2 +1)))dx

$$\int\mathrm{e}^{\mathrm{tan}^{−\mathrm{1}} \mathrm{x}} \left(\frac{\mathrm{1}+\mathrm{x}+\mathrm{x}^{\mathrm{2}} }{\mathrm{x}^{\mathrm{2}} +\mathrm{1}}\right)\mathrm{dx} \\ $$

Question Number 151585 Answers: 0 Comments: 0

∫(dx/(x(√(a^n +x^n ))))

$$\int\frac{\mathrm{dx}}{\mathrm{x}\sqrt{\mathrm{a}^{\mathrm{n}} +\mathrm{x}^{\mathrm{n}} }} \\ $$

Question Number 151630 Answers: 2 Comments: 0

Question Number 151573 Answers: 1 Comments: 0

Question Number 151568 Answers: 4 Comments: 0

∫ (√(sec(x)+tan(x))) dx how can it solve

$$\int\:\sqrt{{sec}\left({x}\right)+{tan}\left({x}\right)}\:{dx} \\ $$$$ \\ $$$${how}\:{can}\:{it}\:{solve} \\ $$

Question Number 151561 Answers: 1 Comments: 0

Given that x+iy=(a/(b+sin θ+icos θ)) show that (b^2 −1)(x^2 +y^2 )+a^2 =2abx

$$\mathrm{Given}\:\mathrm{that}\:\mathrm{x}+\mathrm{iy}=\frac{\mathrm{a}}{\mathrm{b}+\mathrm{sin}\:\theta+\mathrm{icos}\:\theta} \\ $$$$\mathrm{show}\:\mathrm{that} \\ $$$$\left(\mathrm{b}^{\mathrm{2}} −\mathrm{1}\right)\left(\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} \right)+\mathrm{a}^{\mathrm{2}} =\mathrm{2abx} \\ $$

Question Number 151560 Answers: 4 Comments: 0

show that i^i is always real

$$\mathrm{show}\:\mathrm{that}\:\:\mathrm{i}^{\mathrm{i}} \:\:\mathrm{is}\:\mathrm{always}\:\mathrm{real} \\ $$

Question Number 151559 Answers: 2 Comments: 0

prove that (1+cos θ+isin θ)^n + (1+cos θ−isin θ)^n =2^(n+1) cos (θ/2)cos ((nθ)/2)

$$\mathrm{prove}\:\mathrm{that} \\ $$$$\left(\mathrm{1}+\mathrm{cos}\:\theta+\mathrm{isin}\:\theta\right)^{\mathrm{n}} \\ $$$$+\:\left(\mathrm{1}+\mathrm{cos}\:\theta−\mathrm{isin}\:\theta\right)^{\mathrm{n}} =\mathrm{2}^{\mathrm{n}+\mathrm{1}} \mathrm{cos}\:\frac{\theta}{\mathrm{2}}\mathrm{cos}\:\frac{\mathrm{n}\theta}{\mathrm{2}} \\ $$

Question Number 151554 Answers: 1 Comments: 0

Question Number 151549 Answers: 1 Comments: 0

Compare: 2^2^2^.^.^. and 3^3^3^.^.^. Here it is raised 1001 times a square, 1000 times a cube.

$$\mathrm{Compare}: \\ $$$$\mathrm{2}^{\mathrm{2}^{\mathrm{2}^{.^{.^{.} } } } } \:\:\:\:\:\mathrm{and}\:\:\:\:\:\mathrm{3}^{\mathrm{3}^{\mathrm{3}^{.^{.^{.} } } } } \\ $$$$\mathrm{Here}\:\mathrm{it}\:\mathrm{is}\:\mathrm{raised}\:\mathrm{1001}\:\mathrm{times}\:\mathrm{a}\:\mathrm{square}, \\ $$$$\mathrm{1000}\:\mathrm{times}\:\mathrm{a}\:\mathrm{cube}. \\ $$

Question Number 151538 Answers: 3 Comments: 0

Question Number 151533 Answers: 0 Comments: 10

Compare: 2^2^2 and 3^3^3

$$\mathrm{Compare}: \\ $$$$\mathrm{2}^{\mathrm{2}^{\mathrm{2}} } \:\:\:\mathrm{and}\:\:\:\mathrm{3}^{\mathrm{3}^{\mathrm{3}} } \\ $$

Question Number 151531 Answers: 0 Comments: 1

Question Number 151519 Answers: 0 Comments: 0

∫_0 ^( ∞) ((ln(⌊x^2 ⌋!))/( (x^x +1)^x )) dx

$$\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\infty} \:\frac{\mathrm{ln}\left(\lfloor{x}^{\mathrm{2}} \rfloor!\right)}{\:\left({x}^{{x}} +\mathrm{1}\right)^{{x}} }\:{dx} \\ $$$$\: \\ $$

Question Number 151518 Answers: 0 Comments: 0

Question Number 151513 Answers: 2 Comments: 0

Find two possible values of p if the lines px−y=0 and 3x+y+1=0 intersect at 45°

$$\mathrm{Find}\:\mathrm{two}\:\mathrm{possible}\:\mathrm{values}\:\mathrm{of}\:{p}\:\mathrm{if}\:\mathrm{the}\:\mathrm{lines} \\ $$$${px}−{y}=\mathrm{0}\:\mathrm{and}\:\mathrm{3}{x}+{y}+\mathrm{1}=\mathrm{0}\:\mathrm{intersect}\:\mathrm{at}\:\mathrm{45}° \\ $$

Question Number 151504 Answers: 1 Comments: 0

∫_0 ^( ∞) ((ln x)/( (√x) (√(x+1)) (√(2x+1)))) dx

$$\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\infty} \:\frac{\mathrm{ln}\:{x}}{\:\sqrt{{x}}\:\sqrt{{x}+\mathrm{1}}\:\sqrt{\mathrm{2}{x}+\mathrm{1}}}\:{dx} \\ $$$$\: \\ $$

Question Number 151503 Answers: 1 Comments: 0

∫_0 ^( ∞) ((x−1)/( (√(2^x −1)) ln(2^x −1))) dx

$$\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\infty} \:\frac{{x}−\mathrm{1}}{\:\sqrt{\mathrm{2}^{{x}} −\mathrm{1}}\:\mathrm{ln}\left(\mathrm{2}^{{x}} −\mathrm{1}\right)}\:{dx} \\ $$$$\: \\ $$

Question Number 151501 Answers: 1 Comments: 0

∫ (log x + 1)x^x dx

$$\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int\:\left(\mathrm{log}\:{x}\:+\:\mathrm{1}\right){x}^{{x}} \:{dx} \\ $$$$\: \\ $$

Question Number 151496 Answers: 1 Comments: 0

∀x,y∈R_+ ^∗ , show that (x/(x^4 +y^2 ))+(y/(y^4 +x^2 ))≤(1/(xy))

$$\forall{x},{y}\in\mathbb{R}_{+} ^{\ast} ,\:{show}\:{that}\:\frac{{x}}{{x}^{\mathrm{4}} +{y}^{\mathrm{2}} }+\frac{{y}}{{y}^{\mathrm{4}} +{x}^{\mathrm{2}} }\leqslant\frac{\mathrm{1}}{{xy}} \\ $$

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