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AllQuestion and Answers: Page 1556

Question Number 54152 Answers: 0 Comments: 3

Question Number 54137 Answers: 1 Comments: 2

Question Number 54126 Answers: 1 Comments: 0

Prove that ((sin 19θ)/(sin θ)) = cos(−18θ)+cos(−16θ)+... ...+ cos(−2θ)+1+cos(2θ)+....+cos(18θ).

$${Prove}\:{that} \\ $$$$\frac{\mathrm{sin}\:\mathrm{19}\theta}{\mathrm{sin}\:\theta}\:=\:\mathrm{cos}\left(−\mathrm{18}\theta\right)+\mathrm{cos}\left(−\mathrm{16}\theta\right)+... \\ $$$$\:\:\:...+\:\mathrm{cos}\left(−\mathrm{2}\theta\right)+\mathrm{1}+\mathrm{cos}\left(\mathrm{2}\theta\right)+....+\mathrm{cos}\left(\mathrm{18}\theta\right). \\ $$

Question Number 54116 Answers: 0 Comments: 0

Question Number 54102 Answers: 1 Comments: 3

the absolute value ∫_(10) ^(19) ((cos x)/(1+x^8 )) dx is...

$$\mathrm{the}\:\mathrm{absolute}\:\mathrm{value} \\ $$$$\int_{\mathrm{10}} ^{\mathrm{19}} \frac{\mathrm{cos}\:{x}}{\mathrm{1}+{x}^{\mathrm{8}} }\:{dx}\:\mathrm{is}... \\ $$

Question Number 54093 Answers: 0 Comments: 8

If tan(z) = 2, find the value of z

$$\mathrm{If}\:\:\:\:\:\:\mathrm{tan}\left(\mathrm{z}\right)\:\:=\:\:\mathrm{2},\:\:\:\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\:\mathrm{z} \\ $$

Question Number 54074 Answers: 1 Comments: 5

Evaluate : 1) ∫_0 ^( 1) (dx/((√(1+x))+(√(1−x))+2)) 2) ∫_0 ^( 2) ((ln(1+2x))/(1+x^2 )) 3) ∫_0 ^( π) (x/(√(1+sin^3 x)))((3πcosx+4sinx)sin^2 x+4)dx 4) ∫_0 ^( π) ((x^2 cos^2 x−xsinx−cosx−1)/((1+xsinx)^2 )) dx.

$${Evaluate}\:: \\ $$$$\left.\mathrm{1}\right)\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{dx}}{\sqrt{\mathrm{1}+{x}}+\sqrt{\mathrm{1}−{x}}+\mathrm{2}}\: \\ $$$$ \\ $$$$\left.\mathrm{2}\right)\:\:\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{2}} \frac{{ln}\left(\mathrm{1}+\mathrm{2}{x}\right)}{\mathrm{1}+{x}^{\mathrm{2}} } \\ $$$$\left.\mathrm{3}\right) \\ $$$$\:\int_{\mathrm{0}} ^{\:\pi} \frac{{x}}{\sqrt{\mathrm{1}+\mathrm{sin}^{\mathrm{3}} {x}}}\left(\left(\mathrm{3}\pi\mathrm{cos}{x}+\mathrm{4sin}{x}\right)\mathrm{sin}^{\mathrm{2}} {x}+\mathrm{4}\right){dx} \\ $$$$\left.\mathrm{4}\right) \\ $$$$\int_{\mathrm{0}} ^{\:\pi} \:\frac{{x}^{\mathrm{2}} \mathrm{cos}^{\mathrm{2}} {x}−{x}\mathrm{sin}{x}−\mathrm{cos}{x}−\mathrm{1}}{\left(\mathrm{1}+{x}\mathrm{sin}{x}\right)^{\mathrm{2}} }\:{dx}. \\ $$

Question Number 54073 Answers: 1 Comments: 0

Question Number 54070 Answers: 2 Comments: 7

Evaluate : 1) ∫_(−1) ^( 1) cot^(−1) ((1/(√(1−x^2 )))).(cot^(−1) (x/(√(1−(x^2 )^(∣x∣) ))))dx 2) ∫_0 ^( (π/2)) ((sin^2 (10)θ)/(sin^2 θ)) dθ 3) ∫_0 ^( (π/4)) ((ln(cotx))/(((sinx)^(2009) +(cosx)^(2009) )^2 )).(sin2x)^(2008) dx 4) ∫_0 ^( 2) ((4x+10)/((x^2 +5x+6)^2 )) dx.

$${Evaluate}\:: \\ $$$$\left.\mathrm{1}\right) \\ $$$$\:\int_{−\mathrm{1}} ^{\:\mathrm{1}} \mathrm{cot}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}\right).\left(\mathrm{co}{t}^{−\mathrm{1}} \frac{{x}}{\sqrt{\mathrm{1}−\left({x}^{\mathrm{2}} \right)^{\mid{x}\mid} }}\right){dx} \\ $$$$\left.\mathrm{2}\right) \\ $$$$\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \:\frac{\mathrm{sin}^{\mathrm{2}} \left(\mathrm{10}\right)\theta}{\mathrm{sin}^{\mathrm{2}} \theta}\:{d}\theta \\ $$$$\left.\mathrm{3}\right) \\ $$$$\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{4}}} \:\frac{{ln}\left({cotx}\right)}{\left(\left(\mathrm{sin}{x}\right)^{\mathrm{2009}} +\left(\mathrm{cos}{x}\right)^{\mathrm{2009}} \right)^{\mathrm{2}} }.\left(\mathrm{sin2}{x}\right)^{\mathrm{2008}} {dx} \\ $$$$\left.\mathrm{4}\right) \\ $$$$\int_{\mathrm{0}} ^{\:\mathrm{2}} \:\frac{\mathrm{4}{x}+\mathrm{10}}{\left({x}^{\mathrm{2}} +\mathrm{5}{x}+\mathrm{6}\right)^{\mathrm{2}} }\:{dx}. \\ $$

Question Number 54068 Answers: 1 Comments: 2

Question Number 54061 Answers: 0 Comments: 8

Question Number 54051 Answers: 1 Comments: 0

Question Number 54045 Answers: 2 Comments: 1

Question Number 54034 Answers: 0 Comments: 1

If f(x)=f(a+b−x), then ∫_a ^b x f(x) dx =

$$\mathrm{If}\:{f}\left({x}\right)={f}\left({a}+{b}−{x}\right),\:\mathrm{then}\:\underset{{a}} {\overset{{b}} {\int}}\:{x}\:{f}\left({x}\right)\:{dx}\:= \\ $$

Question Number 54033 Answers: 2 Comments: 1

∫_( 0) ^1 (√((1−x)/(1+x))) dx =

$$\underset{\:\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\:\sqrt{\frac{\mathrm{1}−{x}}{\mathrm{1}+{x}}}\:{dx}\:= \\ $$

Question Number 54032 Answers: 0 Comments: 4

∫_(−2 ) ^2 min (x−[x], −x−[−x])dx equals [x] represents greatest integer less than or equal to x).

$$\underset{−\mathrm{2}\:} {\overset{\mathrm{2}} {\int}}\:\mathrm{min}\:\left({x}−\left[{x}\right],\:−{x}−\left[−{x}\right]\right){dx}\:\mathrm{equals}\:\left[{x}\right] \\ $$$$\mathrm{represents}\:\mathrm{greatest}\:\mathrm{integer}\:\mathrm{less}\:\mathrm{than}\:\mathrm{or} \\ $$$$\left.\mathrm{equal}\:\mathrm{to}\:{x}\right). \\ $$

Question Number 54031 Answers: 1 Comments: 0

If f(a+b−x)= f(x), then ∫_a ^b x f(x) dx =

$$\mathrm{If}\:\:{f}\left({a}+{b}−{x}\right)=\:{f}\left({x}\right),\:\mathrm{then}\:\underset{{a}} {\overset{{b}} {\int}}\:{x}\:{f}\left({x}\right)\:{dx}\:= \\ $$

Question Number 54029 Answers: 1 Comments: 1

∫_( 3) ^9 ((√x)/((√(12−x)) + (√x))) dx = 9

$$\underset{\:\mathrm{3}} {\overset{\mathrm{9}} {\int}}\:\frac{\sqrt{{x}}}{\sqrt{\mathrm{12}−{x}}\:+\:\sqrt{{x}}}\:{dx}\:=\:\mathrm{9} \\ $$

Question Number 54028 Answers: 1 Comments: 0

If for every integer n, ∫_n ^(n+1) f(x) dx = n^2 , then the value of ∫_(−2) ^4 f(x) dx=

$$\mathrm{If}\:\mathrm{for}\:\mathrm{every}\:\mathrm{integer}\:{n},\:\underset{{n}} {\overset{{n}+\mathrm{1}} {\int}}{f}\left({x}\right)\:{dx}\:=\:{n}^{\mathrm{2}} , \\ $$$$\mathrm{then}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\underset{−\mathrm{2}} {\overset{\mathrm{4}} {\int}}\:{f}\left({x}\right)\:{dx}= \\ $$

Question Number 54025 Answers: 1 Comments: 1

The points A, B, C, D have coordinates (−7,9), (3,4), (1,12), and (−2,−9). find the length of the linePQ where P devides AB in the ratio 2:3 and devides CD in the ratio 1:−4.

$$\mathrm{The}\:\mathrm{points}\:\mathrm{A},\:\mathrm{B},\:\mathrm{C},\:\mathrm{D}\:\mathrm{have}\:\mathrm{coordinates} \\ $$$$\left(−\mathrm{7},\mathrm{9}\right),\:\left(\mathrm{3},\mathrm{4}\right),\:\left(\mathrm{1},\mathrm{12}\right),\:\mathrm{and}\:\left(−\mathrm{2},−\mathrm{9}\right). \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{length}\:\mathrm{of}\:\mathrm{the}\:\mathrm{linePQ}\:\mathrm{where}\:\mathrm{P} \\ $$$$\mathrm{devides}\:\mathrm{AB}\:\mathrm{in}\:\mathrm{the}\:\mathrm{ratio}\:\mathrm{2}:\mathrm{3}\:\mathrm{and}\:\mathrm{devides} \\ $$$$\mathrm{CD}\:\mathrm{in}\:\mathrm{the}\:\mathrm{ratio}\:\mathrm{1}:−\mathrm{4}. \\ $$

Question Number 54024 Answers: 1 Comments: 0

The coefficient of x^4 in (2−4x+3x^2 )^(−2) i s ?

$$\mathrm{The}\:\mathrm{coefficient}\:\mathrm{of}\:\mathrm{x}^{\mathrm{4}} \:\mathrm{in}\:\left(\mathrm{2}−\mathrm{4x}+\mathrm{3x}^{\mathrm{2}} \right)^{−\mathrm{2}} \mathrm{i}\:\mathrm{s}\:? \\ $$

Question Number 54022 Answers: 0 Comments: 4

Question Number 54015 Answers: 1 Comments: 0

A stone is thrown vertically upwards from the top of a tower 50.0m high with an initial velocity of 20.0ms^(−1) .calculate i.the maximum height the stone reaches ii.the time it takes to reach the maximum height iii.the total distance it covers[take g=10ms^(−1) ]

$$\mathrm{A}\:\mathrm{stone}\:\mathrm{is}\:\mathrm{thrown}\:\mathrm{vertically}\:\mathrm{upwards}\:\mathrm{from}\:\mathrm{the}\:\mathrm{top}\:\mathrm{of}\:\mathrm{a}\:\mathrm{tower}\:\mathrm{50}.\mathrm{0m}\:\mathrm{high}\:\mathrm{with}\:\mathrm{an}\:\mathrm{initial}\:\mathrm{velocity}\:\mathrm{of}\:\mathrm{20}.\mathrm{0ms}^{−\mathrm{1}} .\mathrm{calculate} \\ $$$$\mathrm{i}.\mathrm{the}\:\mathrm{maximum}\:\mathrm{height}\:\mathrm{the}\:\mathrm{stone}\:\mathrm{reaches} \\ $$$$\mathrm{ii}.\mathrm{the}\:\mathrm{time}\:\mathrm{it}\:\mathrm{takes}\:\mathrm{to}\:\mathrm{reach}\:\mathrm{the}\:\mathrm{maximum}\:\mathrm{height} \\ $$$$\mathrm{iii}.\mathrm{the}\:\mathrm{total}\:\mathrm{distance}\:\mathrm{it}\:\mathrm{covers}\left[\mathrm{take}\:\mathrm{g}=\mathrm{10ms}^{−\mathrm{1}} \right] \\ $$

Question Number 54013 Answers: 1 Comments: 0

2^x =−4.solve for x

$$\mathrm{2}^{\boldsymbol{\mathrm{x}}} =−\mathrm{4}.\boldsymbol{\mathrm{solve}}\:\boldsymbol{\mathrm{for}}\:\boldsymbol{\mathrm{x}} \\ $$

Question Number 54011 Answers: 1 Comments: 2

calculate f(a) =∫ (dx/((√(1+ax))−(√(1−ax)))) with a>0 . 2) calculate U_n =∫_(−(1/(na))) ^(1/(na)) (dx/((√(1+ax))−(√(1−ax)))) with n from N and n>1 find lim_(n→+∞) U_n and study the convergence of Σ U_n

$${calculate}\:{f}\left({a}\right)\:=\int\:\:\:\:\frac{{dx}}{\sqrt{\mathrm{1}+{ax}}−\sqrt{\mathrm{1}−{ax}}}\:\:{with}\:{a}>\mathrm{0}\:. \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:{U}_{{n}} =\int_{−\frac{\mathrm{1}}{{na}}} ^{\frac{\mathrm{1}}{{na}}} \:\:\frac{{dx}}{\sqrt{\mathrm{1}+{ax}}−\sqrt{\mathrm{1}−{ax}}}\:\:{with}\:{n}\:{from}\:{N}\:{and}\:{n}>\mathrm{1} \\ $$$${find}\:{lim}_{{n}\rightarrow+\infty} \:{U}_{{n}} \:\:\:{and}\:{study}\:{the}\:{convergence}\:{of}\:\Sigma\:{U}_{{n}} \\ $$

Question Number 53998 Answers: 0 Comments: 1

In the next link, Barry Barrish, who won the physic nobel prize in 2017; gave an exclusive interview to IFT-Instituto de Fisica Teo^ rica-(part 1):

$$\mathrm{In}\:\mathrm{the}\:\mathrm{next}\:\mathrm{link},\:\mathrm{Barry}\:\mathrm{Barrish},\:\mathrm{who}\:\mathrm{won}\:\mathrm{the}\:\mathrm{physic}\:\mathrm{nobel}\:\mathrm{prize}\:\mathrm{in}\:\mathrm{2017}; \\ $$$$\mathrm{gave}\:\mathrm{an}\:\mathrm{exclusive}\:\mathrm{interview}\:\mathrm{to}\:\mathrm{IFT}-{Instituto}\:{de}\:{Fisica}\:{Te}\acute {{o}rica}-\left(\mathrm{part}\:\mathrm{1}\right): \\ $$

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