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AllQuestion and Answers: Page 635

Question Number 152160 Answers: 1 Comments: 0

∫[(1/(log x))−(1/((log x)^2 ))]dx

$$\int\left[\frac{\mathrm{1}}{\mathrm{log}\:\mathrm{x}}−\frac{\mathrm{1}}{\left(\mathrm{log}\:\mathrm{x}\right)^{\mathrm{2}} }\right]\mathrm{dx} \\ $$

Question Number 152197 Answers: 0 Comments: 0

Question Number 152151 Answers: 3 Comments: 0

Question Number 152150 Answers: 1 Comments: 0

Question Number 152147 Answers: 1 Comments: 0

Question Number 152142 Answers: 1 Comments: 0

∫_0 ^1 (x/((1−x+x^2 )^2 ))ln(ln(1/x))dx=−(γ/3)−(1/3)ln((6(√3))/π)+((π(√3))/(27))(5ln2π−6lnΓ((1/6)))

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{x}}{\left(\mathrm{1}−{x}+{x}^{\mathrm{2}} \right)^{\mathrm{2}} }\mathrm{ln}\left(\mathrm{ln}\frac{\mathrm{1}}{{x}}\right){dx}=−\frac{\gamma}{\mathrm{3}}−\frac{\mathrm{1}}{\mathrm{3}}\mathrm{ln}\frac{\mathrm{6}\sqrt{\mathrm{3}}}{\pi}+\frac{\pi\sqrt{\mathrm{3}}}{\mathrm{27}}\left(\mathrm{5ln2}\pi−\mathrm{6ln}\Gamma\left(\frac{\mathrm{1}}{\mathrm{6}}\right)\right) \\ $$

Question Number 152141 Answers: 2 Comments: 0

∫_0 ^(+∞) (((sinx)^(2n+1) )/x)dx=(π/2^(2n+1) ) (((2n)),(n) )

$$\int_{\mathrm{0}} ^{+\infty} \frac{\left(\mathrm{sin}{x}\right)^{\mathrm{2}{n}+\mathrm{1}} }{{x}}{dx}=\frac{\pi}{\mathrm{2}^{\mathrm{2}{n}+\mathrm{1}} }\begin{pmatrix}{\mathrm{2}{n}}\\{{n}}\end{pmatrix} \\ $$

Question Number 152140 Answers: 0 Comments: 0

I=∫_0 ^(2nπ) max(sinx, sin^(−1) (sinx))dx

$${I}=\int_{\mathrm{0}} ^{\mathrm{2n}\pi} \mathrm{max}\left(\mathrm{sin}{x},\:\mathrm{sin}^{−\mathrm{1}} \left(\mathrm{sin}{x}\right)\right){dx} \\ $$

Question Number 152139 Answers: 1 Comments: 1

How many numbers x are found between 9000 and 11000 and verify x=12[19], x=5[13], x=9[11] ?

$$\mathrm{How}\:\mathrm{many}\:\mathrm{numbers}\:{x}\:\mathrm{are}\:\mathrm{found}\:\mathrm{between}\:\mathrm{9000}\:\mathrm{and}\:\mathrm{11000} \\ $$$$\mathrm{and}\:\mathrm{verify}\:{x}=\mathrm{12}\left[\mathrm{19}\right],\:{x}=\mathrm{5}\left[\mathrm{13}\right],\:{x}=\mathrm{9}\left[\mathrm{11}\right]\:? \\ $$

Question Number 152122 Answers: 1 Comments: 0

(√(3+(√(6+(√(9+...+(√(96+(√(99)))))))))) = ?

$$\sqrt{\mathrm{3}+\sqrt{\mathrm{6}+\sqrt{\mathrm{9}+...+\sqrt{\mathrm{96}+\sqrt{\mathrm{99}}}}}}\:=\:? \\ $$$$ \\ $$

Question Number 152115 Answers: 1 Comments: 0

∫(dx/(asin x+bcos x))

$$\int\frac{\mathrm{dx}}{\mathrm{asin}\:\mathrm{x}+\mathrm{bcos}\:\mathrm{x}} \\ $$

Question Number 152116 Answers: 3 Comments: 2

∫((sin x)/( (√(1+sin x))))dx

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

Question Number 152112 Answers: 1 Comments: 0

If I_n =∫((cos nx)/(cos x))dx then 1_n =?

$$\mathrm{If}\:\mathrm{I}_{\mathrm{n}} =\int\frac{\mathrm{cos}\:\mathrm{nx}}{\mathrm{cos}\:\mathrm{x}}\mathrm{dx}\:\:\:\mathrm{then}\:\mathrm{1}_{\mathrm{n}} =? \\ $$

Question Number 152111 Answers: 3 Comments: 0

∫tan^(−1) (sec x+tan x)dx

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

Question Number 152110 Answers: 3 Comments: 0

∫a^(mx) b^(nx) dx

$$\int\mathrm{a}^{\mathrm{mx}} \mathrm{b}^{\mathrm{nx}} \mathrm{dx} \\ $$

Question Number 152106 Answers: 0 Comments: 0

∫ x^n cos(nx) dx

$$\int\:\mathrm{x}^{\mathrm{n}} \:\mathrm{cos}\left(\mathrm{nx}\right)\:\mathrm{dx} \\ $$

Question Number 152094 Answers: 1 Comments: 0

Question Number 152091 Answers: 1 Comments: 0

Given tbat Arg(z+1)=(Π/6) and Arg(z−1)=((2Π)/3).Find z. please help me

$${Given}\:{tbat}\:{Arg}\left({z}+\mathrm{1}\right)=\frac{\Pi}{\mathrm{6}}\:{and}\: \\ $$$${Arg}\left({z}−\mathrm{1}\right)=\frac{\mathrm{2}\Pi}{\mathrm{3}}.{Find}\:{z}. \\ $$$${please}\:{help}\:{me} \\ $$

Question Number 152088 Answers: 2 Comments: 0

∫_0 ^(Π/2) ∣sinx−cosx∣ please help me out

$$\int_{\mathrm{0}} ^{\frac{\Pi}{\mathrm{2}}} \mid{sinx}−{cosx}\mid \\ $$$${please}\:{help}\:{me}\:{out} \\ $$

Question Number 152186 Answers: 1 Comments: 0

∫x^n cos(nx) dx

$$\int\mathrm{x}^{\mathrm{n}} \:\mathrm{cos}\left(\mathrm{nx}\right)\:\mathrm{dx} \\ $$

Question Number 152082 Answers: 1 Comments: 0

Question Number 152101 Answers: 1 Comments: 0

Mr Bonsu an engineer walked round a cylindrical container 4m high once keeping a constant distance of 1m from the container. If he walked with a speed of 3kmh^(−1) for three minutes, calculate to the nearest whole number the: (i) radius of the circle (ii) volume of the container

$$\mathrm{Mr}\:\mathrm{Bonsu}\:\mathrm{an}\:\mathrm{engineer}\:\mathrm{walked}\:\mathrm{round} \\ $$$$\mathrm{a}\:\mathrm{cylindrical}\:\mathrm{container}\:\mathrm{4m}\:\mathrm{high}\:\mathrm{once} \\ $$$$\mathrm{keeping}\:\mathrm{a}\:\mathrm{constant}\:\mathrm{distance}\:\mathrm{of}\:\mathrm{1m}\:\mathrm{from} \\ $$$$\mathrm{the}\:\mathrm{container}.\:\mathrm{If}\:\mathrm{he}\:\mathrm{walked}\:\mathrm{with}\:\mathrm{a}\: \\ $$$$\mathrm{speed}\:\mathrm{of}\:\mathrm{3kmh}^{−\mathrm{1}} \:\mathrm{for}\:\mathrm{three}\:\mathrm{minutes},\: \\ $$$$\mathrm{calculate}\:\mathrm{to}\:\mathrm{the}\:\mathrm{nearest}\:\mathrm{whole}\:\mathrm{number} \\ $$$$\mathrm{the}: \\ $$$$\left(\mathrm{i}\right)\:\mathrm{radius}\:\mathrm{of}\:\mathrm{the}\:\mathrm{circle} \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{volume}\:\mathrm{of}\:\mathrm{the}\:\mathrm{container} \\ $$

Question Number 152063 Answers: 2 Comments: 0

If x is a real number and y is equal to ((x^2 + 1)/(x^2 + x + 1)), show that ∣y − (4/3)∣ ≤ (2/3)

$$\mathrm{If}\:\:\mathrm{x}\:\:\mathrm{is}\:\mathrm{a}\:\mathrm{real}\:\mathrm{number}\:\mathrm{and}\:\:\:\mathrm{y}\:\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to}\:\:\:\:\:\frac{\mathrm{x}^{\mathrm{2}} \:\:+\:\:\mathrm{1}}{\mathrm{x}^{\mathrm{2}} \:\:+\:\:\mathrm{x}\:\:+\:\:\mathrm{1}}, \\ $$$$\mathrm{show}\:\mathrm{that}\:\:\:\:\:\mid\mathrm{y}\:\:\:−\:\:\:\frac{\mathrm{4}}{\mathrm{3}}\mid\:\:\:\leqslant\:\:\:\frac{\mathrm{2}}{\mathrm{3}} \\ $$

Question Number 152061 Answers: 1 Comments: 0

solve Ω= ∫_0 ^( ∞) (( sin^( 3) (x).cos^( 2) (x))/x^( 3) )dx=^? ((7π)/(32))...■

$$ \\ $$$$\:\:\:{solve} \\ $$$$\:\:\:\: \\ $$$$\:\Omega=\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:{sin}^{\:\mathrm{3}} \left({x}\right).{cos}^{\:\mathrm{2}} \left({x}\right)}{{x}^{\:\mathrm{3}} }{dx}\overset{?} {=}\frac{\mathrm{7}\pi}{\mathrm{32}}...\blacksquare \\ $$

Question Number 152052 Answers: 1 Comments: 0

knowns x_2 =(2/x_1 ) , x_3 =(3/x_2 ) , x_4 =(4/x_3 ) , x_5 =(5/x_4 ) , ..., x_8 =(8/x_7 ). Find the value of x_1 ×x_2 ×x_3 ×...×x_8 .

$$\:\:\mathrm{knowns}\:\mathrm{x}_{\mathrm{2}} =\frac{\mathrm{2}}{\mathrm{x}_{\mathrm{1}} }\:,\:\mathrm{x}_{\mathrm{3}} =\frac{\mathrm{3}}{\mathrm{x}_{\mathrm{2}} }\:,\:\mathrm{x}_{\mathrm{4}} =\frac{\mathrm{4}}{\mathrm{x}_{\mathrm{3}} } \\ $$$$,\:\mathrm{x}_{\mathrm{5}} =\frac{\mathrm{5}}{\mathrm{x}_{\mathrm{4}} }\:,\:...,\:\mathrm{x}_{\mathrm{8}} =\frac{\mathrm{8}}{\mathrm{x}_{\mathrm{7}} }.\:\mathrm{Find}\:\mathrm{the} \\ $$$$\mathrm{value}\:\mathrm{of}\:\mathrm{x}_{\mathrm{1}} ×\mathrm{x}_{\mathrm{2}} ×\mathrm{x}_{\mathrm{3}} ×...×\mathrm{x}_{\mathrm{8}} . \\ $$

Question Number 152049 Answers: 0 Comments: 0

If a;b≥1 then prove that: (a+1+((a+1)/a^2 ))^a ∙ (b+1+((b+1)/b^2 ))^b ≥ 2^(2(1+(√(ab))))

$$\mathrm{If}\:\:\mathrm{a};\mathrm{b}\geqslant\mathrm{1}\:\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\left(\mathrm{a}+\mathrm{1}+\frac{\mathrm{a}+\mathrm{1}}{\mathrm{a}^{\mathrm{2}} }\right)^{\boldsymbol{\mathrm{a}}} \centerdot\:\left(\mathrm{b}+\mathrm{1}+\frac{\mathrm{b}+\mathrm{1}}{\mathrm{b}^{\mathrm{2}} }\right)^{\boldsymbol{\mathrm{b}}} \geqslant\:\mathrm{2}^{\mathrm{2}\left(\mathrm{1}+\sqrt{\boldsymbol{\mathrm{ab}}}\right)} \: \\ $$

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