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IntegrationQuestion and Answers: Page 59

Question Number 154142 Answers: 2 Comments: 0

Question Number 154080 Answers: 1 Comments: 0

Ω =∫_0 ^( (π/2)) ln^2 (((1+sin t)/(1−sin t)))dt

$$\:\:\:\:\Omega\:=\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \mathrm{ln}\:^{\mathrm{2}} \left(\frac{\mathrm{1}+\mathrm{sin}\:{t}}{\mathrm{1}−\mathrm{sin}\:{t}}\right){dt} \\ $$

Question Number 154062 Answers: 0 Comments: 0

Question Number 154038 Answers: 0 Comments: 1

monster integral ∫_(−∞) ^( ∞) sin(x^2 )cos(x^3 ) dx

$$\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{monster}\:\mathrm{integral} \\ $$$$\: \\ $$$$\:\:\:\:\:\:\:\:\int_{−\infty} ^{\:\infty} \mathrm{sin}\left({x}^{\mathrm{2}} \right)\mathrm{cos}\left({x}^{\mathrm{3}} \right)\:{dx} \\ $$$$\: \\ $$$$\: \\ $$

Question Number 154037 Answers: 0 Comments: 0

Prove:: Σ_(n=−∞) ^(+∞) arctan (((sinh x)/(cosh n)))=πx

$$\mathrm{Prove}::\:\:\:\underset{\mathrm{n}=−\infty} {\overset{+\infty} {\sum}}\mathrm{arctan}\:\left(\frac{\mathrm{sinh}\:\mathrm{x}}{\mathrm{cosh}\:\mathrm{n}}\right)=\pi\mathrm{x} \\ $$

Question Number 153949 Answers: 1 Comments: 0

∫_0 ^( ∞) sin(x^2 )cos(x^3 )dx

$$\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\infty} \:\mathrm{sin}\left({x}^{\mathrm{2}} \right)\mathrm{cos}\left({x}^{\mathrm{3}} \right){dx} \\ $$$$\: \\ $$

Question Number 153946 Answers: 0 Comments: 0

show whether ∫_0 ^( ∞) sin(x^2 )cos((√x))dx is solvable

$$\: \\ $$$$\:\:\mathrm{show}\:\mathrm{whether} \\ $$$$\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\infty} \:\mathrm{sin}\left({x}^{\mathrm{2}} \right)\mathrm{cos}\left(\sqrt{{x}}\right){dx} \\ $$$$\:\:\mathrm{is}\:\mathrm{solvable} \\ $$$$\: \\ $$

Question Number 153893 Answers: 0 Comments: 0

∫_( 0) ^( ∞) a Π_(p → 1) ^∞ (((p^2 − x^(2n) )/p^2 ))dx, 1 < 2n < n + 1

$$\int_{\:\mathrm{0}} ^{\:\:\infty} \mathrm{a}\:\underset{\mathrm{p}\:\rightarrow\:\mathrm{1}} {\overset{\infty} {\prod}}\left(\frac{\mathrm{p}^{\mathrm{2}} \:\:−\:\:\:\mathrm{x}^{\mathrm{2n}} }{\mathrm{p}^{\mathrm{2}} }\right)\mathrm{dx},\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}\:\:<\:\:\mathrm{2n}\:\:<\:\:\mathrm{n}\:\:+\:\:\mathrm{1} \\ $$

Question Number 153875 Answers: 0 Comments: 0

Prove that.. 𝛗 : =∫_( 1) ^( +∞) (( ln (x ))/(( x^( π) −1 )( ln^( 2) (x) +1 )^2 ))dx= ((π^( 2) − 8)/(16)) ■

$$ \\ $$$$\:\:\:\:\mathrm{Prove}\:\:\mathrm{that}.. \\ $$$$\:\:\: \\ $$$$\:\:\:\:\boldsymbol{\phi}\::\:=\int_{\:\mathrm{1}} ^{\:+\infty} \frac{\:{ln}\:\left({x}\:\right)}{\left(\:{x}^{\:\pi} \:−\mathrm{1}\:\right)\left(\:{ln}^{\:\mathrm{2}} \left({x}\right)\:+\mathrm{1}\:\right)^{\mathrm{2}} }{dx}=\:\frac{\pi^{\:\mathrm{2}} −\:\mathrm{8}}{\mathrm{16}}\:\:\:\:\:\:\:\:\:\blacksquare\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\: \\ $$

Question Number 153873 Answers: 0 Comments: 1

∫_0 ^( ∞) (( x)/((1 +x^( 2) ) ( e^( 2πx) − 1))) dx =((2γ− 1)/4)

$$ \\ $$$$\int_{\mathrm{0}} ^{\:\infty} \frac{\:{x}}{\left(\mathrm{1}\:+{x}^{\:\mathrm{2}} \right)\:\left(\:{e}^{\:\mathrm{2}\pi{x}} −\:\mathrm{1}\right)}\:{dx}\:=\frac{\mathrm{2}\gamma−\:\mathrm{1}}{\mathrm{4}} \\ $$$$ \\ $$

Question Number 153759 Answers: 0 Comments: 0

Ω= Σ_(n=1) ^∞ {n^2 (∫_0 ^( (π/2)) (( sin^( 2) (x ))/((sin(x)+cos(x))^( 4) )))^( n) dx}=?

$$ \\ $$$$\Omega=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left\{{n}^{\mathrm{2}} \left(\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \frac{\:{sin}^{\:\mathrm{2}} \left({x}\:\right)}{\left({sin}\left({x}\right)+{cos}\left({x}\right)\right)^{\:\mathrm{4}} }\right)^{\:{n}} {dx}\right\}=? \\ $$$$ \\ $$

Question Number 153734 Answers: 0 Comments: 0

Question Number 153721 Answers: 2 Comments: 0

prove that : I:= ∫_0 ^( ∞) (( x^( 3) )/(sinh ( x ))) dx = ((π^4 )/8) ■ m.n

$$ \\ $$$$\:\:\:\:\:\mathrm{prove}\:\:\mathrm{that}\:: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{I}:=\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:{x}^{\:\mathrm{3}} }{{sinh}\:\left(\:{x}\:\right)}\:{dx}\:=\:\frac{\pi\:^{\mathrm{4}} }{\mathrm{8}}\:\:\:\:\:\:\:\:\:\:\blacksquare\:{m}.{n}\:\:\:\:\:\:\:\: \\ $$$$ \\ $$

Question Number 153555 Answers: 1 Comments: 0

Ω := ∫_0 ^( (π/2)) cos(2x).ln(sin(x))dx=^? −(π/4) solution (1 ) Ω := ∫_0 ^( (π/2)) ( 2cos^( 2) (x)−1)ln(sin(x))dx := 2∫_0 ^( (π/2)) cos^( 2) (x).ln(sin(x))dx−∫_0 ^( (π/2)) ln(sin(x))dx we know that : ∫_0 ^(π/2) ln(sin(x))dx=_(earlier) ^(derived) ((−π)/2) ln(2) ∫_0 ^( (π/2)) cos^( 2) (x).ln(sin(x))dx=_(posts) ^(previous) −(π/4)ln(2)−(π/8) ∴ Ω := −(π/2) ln(2) −(π/4) +(π/2) ln(2) ◂ Ω =− (π/4) ▶ m.n

$$ \\ $$$$\:\:\:\:\:\Omega\::=\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} {cos}\left(\mathrm{2}{x}\right).{ln}\left({sin}\left({x}\right)\right){dx}\overset{?} {=}\:−\frac{\pi}{\mathrm{4}} \\ $$$$\:\:\:\:\:\:\:\:\:\:{solution}\:\left(\mathrm{1}\:\right) \\ $$$$\:\:\:\:\:\Omega\::=\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \left(\:\mathrm{2}{cos}^{\:\mathrm{2}} \left({x}\right)−\mathrm{1}\right){ln}\left({sin}\left({x}\right)\right){dx} \\ $$$$\:\:\:\:\:\::=\:\mathrm{2}\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} {cos}^{\:\mathrm{2}} \left({x}\right).{ln}\left({sin}\left({x}\right)\right){dx}−\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} {ln}\left({sin}\left({x}\right)\right){dx} \\ $$$$\:\:\:\:\:{we}\:{know}\:{that}\::\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {ln}\left({sin}\left({x}\right)\right){dx}\underset{{earlier}} {\overset{{derived}} {=}}\:\frac{−\pi}{\mathrm{2}}\:{ln}\left(\mathrm{2}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} {cos}^{\:\mathrm{2}} \left({x}\right).{ln}\left({sin}\left({x}\right)\right){dx}\underset{{posts}} {\overset{{previous}} {=}}\:−\frac{\pi}{\mathrm{4}}{ln}\left(\mathrm{2}\right)−\frac{\pi}{\mathrm{8}} \\ $$$$\:\:\:\:\:\:\therefore\:\:\:\Omega\::=\:−\frac{\pi}{\mathrm{2}}\:{ln}\left(\mathrm{2}\right)\:−\frac{\pi}{\mathrm{4}}\:+\frac{\pi}{\mathrm{2}}\:{ln}\left(\mathrm{2}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\blacktriangleleft\:\:\:\:\Omega\:=−\:\frac{\pi}{\mathrm{4}}\:\:\blacktriangleright\:\:\:\:\:\:{m}.{n} \\ $$

Question Number 153535 Answers: 1 Comments: 0

find ∫((√(x^2 −9))/x^3 ) dx=?

$${find}\:\int\frac{\sqrt{{x}^{\mathrm{2}} −\mathrm{9}}}{{x}^{\mathrm{3}} }\:{dx}=? \\ $$

Question Number 153430 Answers: 1 Comments: 2