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IntegrationQuestion and Answers: Page 277

Question Number 42086 Answers: 1 Comments: 0

let f(x) =∫_0 ^2 ((ch(t))/(2xsh(t) +1)) dt 1) find a simple form of f(x) 2) calculate ∫_0 ^2 ((ch(t))/(1+sh(t)))dt 3) calculate ∫_0 ^2 ((ch(t))/(3sh(t) +1))dt .

$${let}\:\:\:\:{f}\left({x}\right)\:\:=\int_{\mathrm{0}} ^{\mathrm{2}} \:\:\:\:\:\frac{{ch}\left({t}\right)}{\mathrm{2}{xsh}\left({t}\right)\:+\mathrm{1}}\:{dt} \\ $$$$\left.\mathrm{1}\right)\:{find}\:\:{a}\:{simple}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:\:{calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{2}} \:\:\:\:\frac{{ch}\left({t}\right)}{\mathrm{1}+{sh}\left({t}\right)}{dt} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{2}} \:\:\:\:\frac{{ch}\left({t}\right)}{\mathrm{3}{sh}\left({t}\right)\:+\mathrm{1}}{dt}\:. \\ $$

Question Number 42085 Answers: 1 Comments: 1

calculate ∫_1 ^(+∞) ((2x+1)/(3 +(x+1)^3 ))dx

$${calculate}\:\:\:\:\int_{\mathrm{1}} ^{+\infty} \:\:\:\frac{\mathrm{2}{x}+\mathrm{1}}{\mathrm{3}\:+\left({x}+\mathrm{1}\right)^{\mathrm{3}} }{dx}\: \\ $$

Question Number 42083 Answers: 0 Comments: 0

calculate ∫_0 ^1 ((2x+1)/(3+(1+x)^3 ))dx

$${calculate}\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\:\:\frac{\mathrm{2}{x}+\mathrm{1}}{\mathrm{3}+\left(\mathrm{1}+{x}\right)^{\mathrm{3}} }{dx} \\ $$

Question Number 42054 Answers: 1 Comments: 1

Question Number 42012 Answers: 2 Comments: 0

Solve : (du/dt) = ((3u−7t)/(−7u+3t))

$$\mathrm{Solve}\:: \\ $$$$\frac{{d}\mathrm{u}}{{d}\mathrm{t}}\:=\:\frac{\mathrm{3u}−\mathrm{7t}}{−\mathrm{7u}+\mathrm{3t}} \\ $$

Question Number 41989 Answers: 2 Comments: 1

Question Number 41913 Answers: 1 Comments: 0

let f(a) = ∫_0 ^π (x/(1+acosx))dx 1) find f(a) 2) calculate ∫_0 ^π (x/(1+2cosx))dx and ∫_0 ^π (x/(1−2cosx))dx 3) calculate ∫_0 ^π ((xcosx)/((1+acosx)^2 ))dx 4) find the value of ∫_0 ^π ((xcosx)/((1+2cosx)^2 ))dx .

$${let}\:{f}\left({a}\right)\:=\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\frac{{x}}{\mathrm{1}+{acosx}}{dx} \\ $$$$\left.\mathrm{1}\right)\:{find}\:\:{f}\left({a}\right)\: \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{x}}{\mathrm{1}+\mathrm{2}{cosx}}{dx}\:{and}\:\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{x}}{\mathrm{1}−\mathrm{2}{cosx}}{dx} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{\pi} \:\:\frac{{xcosx}}{\left(\mathrm{1}+{acosx}\right)^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{4}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{xcosx}}{\left(\mathrm{1}+\mathrm{2}{cosx}\right)^{\mathrm{2}} }{dx}\:. \\ $$

Question Number 41854 Answers: 0 Comments: 0

1)calculate ∫_0 ^1 ln(1+x+x^2 +x^3 )dx 2)then find the value of ∫_0 ^1 ln( 1−x^5 ) dx 3) find the value of Σ_(n=1) ^∞ (1/(n(5n+1))) .

$$\left.\mathrm{1}\right){calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}+{x}+{x}^{\mathrm{2}} \:+{x}^{\mathrm{3}} \right){dx} \\ $$$$\left.\mathrm{2}\right){then}\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{ln}\left(\:\mathrm{1}−{x}^{\mathrm{5}} \right)\:{dx} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{value}\:{of}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\frac{\mathrm{1}}{{n}\left(\mathrm{5}{n}+\mathrm{1}\right)}\:. \\ $$$$ \\ $$$$ \\ $$

Question Number 41851 Answers: 0 Comments: 2

Question Number 41847 Answers: 1 Comments: 0

let f(x) = ∫_0 ^(π/4) (dt/(x +tan(t))) 1) find anoher expression off (x) 2) calculate ∫_0 ^(π/4) (dt/(2+tan(t))) and A(θ) = ∫_0 ^(π/4) (dt/(sinθ+tant)) 3) calculate ∫_0 ^(π/4) (dt/((1+tant)^2 ))

$${let}\:{f}\left({x}\right)\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\:\:\:\frac{{dt}}{{x}\:+{tan}\left({t}\right)} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{anoher}\:{expression}\:{off}\:\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\frac{{dt}}{\mathrm{2}+{tan}\left({t}\right)}\:\:\:{and}\:\:{A}\left(\theta\right)\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\:\:\frac{{dt}}{{sin}\theta+{tant}} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\:\:\frac{{dt}}{\left(\mathrm{1}+{tant}\right)^{\mathrm{2}} } \\ $$

Question Number 41846 Answers: 1 Comments: 0

find ∫ (dx/((√(1+x^2 )) +(√(1−x^2 ))))

$${find}\:\:\int\:\:\:\:\:\frac{{dx}}{\sqrt{\mathrm{1}+{x}^{\mathrm{2}} \:}\:\:+\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }} \\ $$

Question Number 41845 Answers: 1 Comments: 0

1)find ∫ (x/((√(1+x)) +(√(1−x)))) dx 2) calculate ∫_1 ^3 (x/((√(1+x)) +(√(1−x)))) dx

$$\left.\mathrm{1}\right){find}\:\:\:\:\int\:\:\:\:\:\:\:\:\:\frac{{x}}{\sqrt{\mathrm{1}+{x}}\:+\sqrt{\mathrm{1}−{x}}}\:{dx} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\int_{\mathrm{1}} ^{\mathrm{3}} \:\:\:\:\:\frac{{x}}{\sqrt{\mathrm{1}+{x}}\:+\sqrt{\mathrm{1}−{x}}}\:{dx} \\ $$

Question Number 41848 Answers: 0 Comments: 3

let f(a) = ∫_0 ^(π/2) (dx/(1+asinx)) with a∈R 1) find a simple form of f(a) 2) calculate ∫_0 ^(π/2) (dx/(1+sinx)) and ∫_0 ^(π/2) (dx/(1+2sinx)) 3) find the value of ∫_0 ^(π/2) ((cosx)/((1+asinx)^2 ))dx 4) find the value of ∫_0 ^(π/2) ((cosx)/((1+sinx)^2 ))dx and ∫_0 ^(π/2) ((cosx)/((1+2sinx)^2 ))dx

$${let}\:{f}\left({a}\right)\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\:\frac{{dx}}{\mathrm{1}+{asinx}}\:\:\:{with}\:{a}\in{R} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{simple}\:{form}\:{of}\:{f}\left({a}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{dx}}{\mathrm{1}+{sinx}}\:{and}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{dx}}{\mathrm{1}+\mathrm{2}{sinx}} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{cosx}}{\left(\mathrm{1}+{asinx}\right)^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{4}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{cosx}}{\left(\mathrm{1}+{sinx}\right)^{\mathrm{2}} }{dx}\:{and}\:\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{cosx}}{\left(\mathrm{1}+\mathrm{2}{sinx}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 41806 Answers: 1 Comments: 1

Question Number 41762 Answers: 0 Comments: 3

let f(x) = ∫_0 ^1 ((ln(1+xt^2 ))/(1+t^2 )) dt 1) find a simple form of f(x) 2) calculate ∫_0 ^1 ((ln(1+t^2 ))/(1+t^2 ))dt 3) calculate ∫_0 ^1 ((ln(1+2t^2 ))/(1+t^2 )) dt

$${let}\:{f}\left({x}\right)\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{ln}\left(\mathrm{1}+{xt}^{\mathrm{2}} \right)}{\mathrm{1}+{t}^{\mathrm{2}} }\:{dt} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{simple}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{ln}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{ln}\left(\mathrm{1}+\mathrm{2}{t}^{\mathrm{2}} \right)}{\mathrm{1}+{t}^{\mathrm{2}} }\:{dt} \\ $$

Question Number 41703 Answers: 0 Comments: 1

calculate I = ∫_0 ^(π/4) ((cosx)/(cos^3 x +sin^3 x))dx

$${calculate}\:\:{I}\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\:\frac{{cosx}}{{cos}^{\mathrm{3}} {x}\:+{sin}^{\mathrm{3}} {x}}{dx} \\ $$

Question Number 41702 Answers: 1 Comments: 3

find the value of ∫_0 ^(√3) arcsin(((2t)/(1+t^2 )))dt

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\sqrt{\mathrm{3}}} \:{arcsin}\left(\frac{\mathrm{2}{t}}{\mathrm{1}+{t}^{\mathrm{2}} }\right){dt} \\ $$

Question Number 41679 Answers: 1 Comments: 5

let f(x) = ∫_0 ^1 ln(1+t +xt^2 )dt 1) calculate f^′ (x) then find a simple form of f(x) 2) calculate ∫_0 ^1 ln(1+t +t^2 )dt 3) calculate ∫_0 ^1 ln(1−t^3 )dt .

$${let}\:{f}\left({x}\right)\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}+{t}\:+{xt}^{\mathrm{2}} \right){dt} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{'} \left({x}\right)\:{then}\:{find}\:{a}\:{simple}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}+{t}\:+{t}^{\mathrm{2}} \right){dt} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:{ln}\left(\mathrm{1}−{t}^{\mathrm{3}} \right){dt}\:. \\ $$

Question Number 41678 Answers: 1 Comments: 0

prove that ∫_0 ^∞ cos(x^2 )dx=∫_0 ^∞ sin(x^2 )dx by using only series.

$${prove}\:{that}\:\:\int_{\mathrm{0}} ^{\infty} \:{cos}\left({x}^{\mathrm{2}} \right){dx}=\int_{\mathrm{0}} ^{\infty} \:{sin}\left({x}^{\mathrm{2}} \right){dx}\:{by}\:{using} \\ $$$${only}\:{series}. \\ $$

Question Number 41677 Answers: 2 Comments: 2

calculate A = ∫_0 ^(π/4) cos^8 xdx and B= ∫_0 ^(π/4) sin^8 xdx 2) calculate A +B and A−B 3) calculate A^2 −B^2

$${calculate}\:{A}\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:{cos}^{\mathrm{8}} {xdx}\:{and}\: \\ $$$${B}=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:{sin}^{\mathrm{8}} {xdx} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{A}\:+{B}\:{and}\:{A}−{B} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:{A}^{\mathrm{2}} \:−{B}^{\mathrm{2}} \\ $$

Question Number 41675 Answers: 1 Comments: 1