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IntegrationQuestion and Answers: Page 51

Question Number 158142 Answers: 0 Comments: 0

f(x)=x−[x] where [x] is the greatest integer function and −3≤x≤3 a) sketch f(x) b) state the domain of f(x) c) study the continuity of f(x) on its domain d) state the range of f(x)

$${f}\left({x}\right)={x}−\left[{x}\right]\:{where}\:\left[{x}\right]\:{is}\:{the}\:{greatest} \\ $$$${integer}\:{function}\:{and}\:−\mathrm{3}\leqslant{x}\leqslant\mathrm{3} \\ $$$$\left.{a}\right)\:{sketch}\:{f}\left({x}\right) \\ $$$$\left.{b}\right)\:{state}\:{the}\:{domain}\:{of}\:{f}\left({x}\right) \\ $$$$\left.{c}\right)\:{study}\:{the}\:{continuity}\:{of}\:{f}\left({x}\right)\:{on}\:{its}\:{domain} \\ $$$$\left.{d}\right)\:{state}\:{the}\:{range}\:{of}\:{f}\left({x}\right) \\ $$

Question Number 158159 Answers: 2 Comments: 0

∫ (dx/(3−tan x)) =?

$$\:\:\:\:\:\:\int\:\frac{{dx}}{\mathrm{3}−\mathrm{tan}\:{x}}\:=? \\ $$

Question Number 158053 Answers: 2 Comments: 0

∫(dx/(sin^4 x))

$$\int\frac{{dx}}{\mathrm{sin}^{\mathrm{4}} {x}} \\ $$

Question Number 158009 Answers: 2 Comments: 0

$$ \\ $$$$ \\ $$

Question Number 157977 Answers: 1 Comments: 0

∫ (dx/((1+(x)^(1/4) )(√x))) =?

$$\int\:\frac{{dx}}{\left(\mathrm{1}+\sqrt[{\mathrm{4}}]{{x}}\:\right)\sqrt{{x}}}\:=? \\ $$

Question Number 157961 Answers: 2 Comments: 0

prove that: I=∫_0 ^( ∞) x^( 2) tanh(x).e^( −x) dx=(π^( 3) /8) −2

$$ \\ $$$$\:\:\:\:\:\:\:{prove}\:{that}: \\ $$$$ \\ $$$$\mathrm{I}=\int_{\mathrm{0}} ^{\:\infty} {x}^{\:\mathrm{2}} {tanh}\left({x}\right).{e}^{\:−{x}} {dx}=\frac{\pi^{\:\mathrm{3}} }{\mathrm{8}}\:−\mathrm{2}\:\:\:\:\:\:\: \\ $$$$ \\ $$

Question Number 157932 Answers: 0 Comments: 4

prove that tan^(−1) (((xy)/(rz)))+tan^(−1) (((xz)/(ry)))+tan^(−1) (((yz)/(rx)))=(π/2)

$${prove}\:{that}\:\mathrm{tan}^{−\mathrm{1}} \left(\frac{{xy}}{{rz}}\right)+\mathrm{tan}^{−\mathrm{1}} \left(\frac{{xz}}{{ry}}\right)+\mathrm{tan}^{−\mathrm{1}} \left(\frac{{yz}}{{rx}}\right)=\frac{\pi}{\mathrm{2}} \\ $$

Question Number 157929 Answers: 0 Comments: 1

∫(dx/(sin x+ sec x)) using wiestress substitution

$$\int\frac{\mathrm{dx}}{\mathrm{sin}\:\mathrm{x}+\:\mathrm{sec}\:\mathrm{x}} \\ $$$$\mathrm{using}\:\mathrm{wiestress}\:\mathrm{substitution} \\ $$

Question Number 157870 Answers: 1 Comments: 0

find the integral: ∫{(3x+1)/(x^2 +4)}dx

$${find}\:{the}\:{integral}: \\ $$$$\int\left\{\left(\mathrm{3}{x}+\mathrm{1}\right)/\left({x}^{\mathrm{2}} +\mathrm{4}\right)\right\}{dx} \\ $$

Question Number 157826 Answers: 1 Comments: 0

Question Number 157823 Answers: 1 Comments: 0

calculate : Ω := Σ_(n=0) ^∞ (( 1)/((4n+1)^( 3) )) = ?

$$ \\ $$$$\:\:\:\:\:\:\:{calculate}\:: \\ $$$$\:\:\:\:\Omega\::=\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\:\mathrm{1}}{\left(\mathrm{4}{n}+\mathrm{1}\right)^{\:\mathrm{3}} }\:\:\:=\:? \\ $$$$\: \\ $$

Question Number 157744 Answers: 1 Comments: 0

∫_0 ^( 1) (( Li_( 2) (−x ) )/(1+ x))dx=?

$$ \\ $$$$\:\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:\:\mathrm{Li}_{\:\mathrm{2}} \:\left(−\mathrm{x}\:\right)\:}{\mathrm{1}+\:\mathrm{x}}\mathrm{dx}=? \\ $$$$ \\ $$$$ \\ $$

Question Number 157750 Answers: 1 Comments: 0

∫_0 ^∞ ((1−cos 4x)/(xe^x )) dx=?

$$\:\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{1}−\mathrm{cos}\:\mathrm{4}{x}}{{xe}^{{x}} }\:{dx}=? \\ $$

Question Number 157655 Answers: 1 Comments: 1

x^2 f(x^3 )+(1/((1+x)^2 )) f(((1−x)/(1+x)))=4x^3 (1+x^4 )^5 ∫_( 0) ^( 1) f(x) dx =?

$$\:\:{x}^{\mathrm{2}} \:{f}\left({x}^{\mathrm{3}} \right)+\frac{\mathrm{1}}{\left(\mathrm{1}+{x}\right)^{\mathrm{2}} }\:{f}\left(\frac{\mathrm{1}−{x}}{\mathrm{1}+{x}}\right)=\mathrm{4}{x}^{\mathrm{3}} \left(\mathrm{1}+{x}^{\mathrm{4}} \right)^{\mathrm{5}} \\ $$$$\:\int_{\:\mathrm{0}} ^{\:\mathrm{1}} {f}\left({x}\right)\:{dx}\:=? \\ $$

Question Number 157531 Answers: 2 Comments: 0

Question Number 157515 Answers: 0 Comments: 0

∫ ((x sin x)/(16x+9)) dx

$$\:\:\:\:\:\int\:\frac{{x}\:\mathrm{sin}\:{x}}{\mathrm{16}{x}+\mathrm{9}}\:{dx}\: \\ $$

Question Number 157469 Answers: 1 Comments: 0

calculate : Ω:= ∫_(0 ) ^( 1) (( ln(1+x).ln(1−x))/(1+x)) dx =?

$$ \\ $$$$\:\:\:\:{calculate}\:: \\ $$$$\:\Omega:=\:\int_{\mathrm{0}\:} ^{\:\mathrm{1}} \frac{\:{ln}\left(\mathrm{1}+{x}\right).{ln}\left(\mathrm{1}−{x}\right)}{\mathrm{1}+{x}}\:{dx}\:=? \\ $$$$\:\:\:\: \\ $$

Question Number 157456 Answers: 4 Comments: 0

∫ 5^(3−2x) dx =?

$$\:\int\:\mathrm{5}^{\mathrm{3}−\mathrm{2}{x}} \:{dx}\:=? \\ $$

Question Number 157455 Answers: 1 Comments: 0

Question Number 157442 Answers: 2 Comments: 0

∫ (dx/(x−(√(x^2 +2x+2))))

$$\:\:\int\:\frac{\mathrm{dx}}{\mathrm{x}−\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{2x}+\mathrm{2}}}\: \\ $$

Question Number 157412 Answers: 2 Comments: 0

∫_( 0) ^( (π/2)) (x/(sin^8 x+cos^8 x)) dx ?

$$\:\int_{\:\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \:\frac{{x}}{\mathrm{sin}\:^{\mathrm{8}} {x}+\mathrm{cos}\:^{\mathrm{8}} {x}}\:{dx}\:? \\ $$

Question Number 157309 Answers: 3 Comments: 0

∫ (dx/( (√x)+x(√(x+1)))) =?

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

Question Number 157268 Answers: 1 Comments: 0

prove that ∫(x^2 /((xsin x+cos x)^2 ))dx=−((xsec x)/(xsin x+cos x))+tan x+c

$${prove}\:{that} \\ $$$$\int\frac{{x}^{\mathrm{2}} }{\left({x}\mathrm{sin}\:{x}+\mathrm{cos}\:{x}\right)^{\mathrm{2}} }{dx}=−\frac{{x}\mathrm{sec}\:{x}}{{x}\mathrm{sin}\:{x}+\mathrm{cos}\:{x}}+\mathrm{tan}\:{x}+{c} \\ $$

Question Number 157257 Answers: 1 Comments: 0

∫ ((sin^6 x+cos^5 x)/(sin^2 x cos^2 x)) dx

$$\int\:\frac{\mathrm{sin}\:^{\mathrm{6}} {x}+\mathrm{cos}\:^{\mathrm{5}} {x}}{\mathrm{sin}\:^{\mathrm{2}} {x}\:\mathrm{cos}\:^{\mathrm{2}} {x}}\:{dx} \\ $$

Question Number 157251 Answers: 1 Comments: 0

# Nice Mathematics # ...calculation ... Ω :=∫_0 ^( 1) (( tanh^( −1) ((√( x)) ))/x) dx =^? (( π^( 2) )/4) −−−−−−−−−−−−− Ω :=^((√x) = t) 2∫_0 ^( 1) (( tanh^( −1) (t ))/t) dt :=^({tanh^( −1) (t )= (1/2) ln( ((1+t)/(1−t)) ) }) ∫_0 ^( 1) ((ln( 1+t )− ln(1−t ))/t) dt : = −Li_( 2) (−1 ) + Li_( 2) (1 ) :=^( {Li_( 2) (z )= Σ_(n=1) ^∞ (( z^( n) )/n^( 2) ) }) η (2) + ζ (2) := (π^( 2) /(12)) + (π^( 2) /6) = (( π^( 2) )/( 4)) ■ m.n

$$ \\ $$$$\:\:\:\:\:#\:\mathrm{Nice}\:\mathrm{Mathematics}\:# \\ $$$$\:\:\:\:\:\:\:...{calculation}\:... \\ $$$$\:\:\:\:\:\:\:\:\Omega\::=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:{tanh}^{\:−\mathrm{1}} \:\left(\sqrt{\:{x}}\:\right)}{{x}}\:{dx}\:\overset{?} {=}\:\frac{\:\pi^{\:\mathrm{2}} }{\mathrm{4}} \\ $$$$\:\:\:−−−−−−−−−−−−− \\ $$$$\:\:\:\:\Omega\::\overset{\sqrt{{x}}\:=\:{t}} {=}\:\mathrm{2}\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:{tanh}^{\:−\mathrm{1}} \:\left({t}\:\right)}{{t}}\:{dt} \\ $$$$\:\:\:\:\:\:\:\:\::\overset{\left\{{tanh}^{\:−\mathrm{1}} \:\left({t}\:\right)=\:\frac{\mathrm{1}}{\mathrm{2}}\:{ln}\left(\:\frac{\mathrm{1}+{t}}{\mathrm{1}−{t}}\:\right)\:\right\}} {=}\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{ln}\left(\:\mathrm{1}+{t}\:\right)−\:{ln}\left(\mathrm{1}−{t}\:\right)}{{t}}\:{dt} \\ $$$$\:\:\:\::\:\:=\:\:−\mathrm{Li}_{\:\mathrm{2}} \:\left(−\mathrm{1}\:\right)\:+\:\mathrm{Li}_{\:\mathrm{2}} \:\left(\mathrm{1}\:\right) \\ $$$$\:\:\:\:\::\overset{\:\left\{\mathrm{Li}_{\:\mathrm{2}} \:\left({z}\:\right)=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\:{z}^{\:{n}} }{{n}^{\:\mathrm{2}} }\:\right\}} {=}\:\:\eta\:\left(\mathrm{2}\right)\:+\:\zeta\:\left(\mathrm{2}\right)\: \\ $$$$\:\:\:\:\::=\:\:\frac{\pi^{\:\mathrm{2}} }{\mathrm{12}}\:+\:\frac{\pi^{\:\mathrm{2}} }{\mathrm{6}}\:\:=\:\frac{\:\pi^{\:\mathrm{2}} }{\:\mathrm{4}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\blacksquare\:{m}.{n}\:\:\:\:\:\: \\ $$$$ \\ $$

Question Number 157254 Answers: 1 Comments: 0

Show that ∫_0 ^1 (1/((x+1)(x+2)))dx = ln((4/3))

$$\mathrm{Show}\:\mathrm{that}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{\mathrm{1}}{\left({x}+\mathrm{1}\right)\left({x}+\mathrm{2}\right)}{dx}\:=\:\mathrm{ln}\left(\frac{\mathrm{4}}{\mathrm{3}}\right) \\ $$

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