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IntegrationQuestion and Answers: Page 269

Question Number 44470 Answers: 0 Comments: 0

find ∫_0 ^∞ (dt/(1+t^2 sin^2 t))

$${find}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dt}}{\mathrm{1}+{t}^{\mathrm{2}} {sin}^{\mathrm{2}} {t}} \\ $$

Question Number 44466 Answers: 0 Comments: 4

let f(x) = ∫_0 ^∞ ((x sinx)/(a^2 +x^4 ))dx with a>0 1) find a explicit form of f(a) 2) find g(a) = ∫_0 ^∞ ((xsinx)/((a^2 +x^4 )^2 ))dx 3)find the value of ∫_0 ^∞ ((x sinx)/(x^4 +1))dx 4) find the value of ∫_0 ^∞ ((xsinx)/((x^4 +1)^2 ))dx .

$${let}\:{f}\left({x}\right)\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{x}\:{sinx}}{{a}^{\mathrm{2}} \:+{x}^{\mathrm{4}} }{dx}\:\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({a}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:\:{g}\left({a}\right)\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{xsinx}}{\left({a}^{\mathrm{2}} \:+{x}^{\mathrm{4}} \right)^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{3}\right){find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{x}\:{sinx}}{{x}^{\mathrm{4}} \:+\mathrm{1}}{dx} \\ $$$$\left.\mathrm{4}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{xsinx}}{\left({x}^{\mathrm{4}} \:+\mathrm{1}\right)^{\mathrm{2}} }{dx}\:. \\ $$$$ \\ $$

Question Number 44441 Answers: 1 Comments: 0

Question Number 44424 Answers: 1 Comments: 0

by considering a sermicircle from −r to r prove that area of circle is πr^2

$${by}\:{considering}\:\:{a}\:{sermicircle}\:{from}\:−{r}\:{to}\:\:{r}\:{prove}\:{that}\:{area}\:{of}\:{circle}\:{is}\:\pi{r}^{\mathrm{2}} \\ $$

Question Number 44423 Answers: 1 Comments: 0

evaluate ∫3^x dx

$${evaluate}\:\int\mathrm{3}^{{x}} {dx} \\ $$

Question Number 44422 Answers: 0 Comments: 3

use substitution x=cos^2 θ+3sin^2 θ show that∫_1 ^3 (dx/(√((x−1)(3−x))))=π

$${use}\:{substitution}\:{x}=\mathrm{cos}\:^{\mathrm{2}} \theta+\mathrm{3}{sin}^{\mathrm{2}} \theta \\ $$$${show}\:{that}\int_{\mathrm{1}} ^{\mathrm{3}} \frac{{dx}}{\sqrt{\left({x}−\mathrm{1}\right)\left(\mathrm{3}−{x}\right)}}=\pi \\ $$

Question Number 44319 Answers: 1 Comments: 2

find lim_(x→0^+ ) ∫_x ^(2x) ((√(1+t^2 ))/t)dt .

$${find}\:{lim}_{{x}\rightarrow\mathrm{0}^{+} } \:\:\:\int_{{x}} ^{\mathrm{2}{x}} \:\frac{\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }}{{t}}{dt}\:. \\ $$

Question Number 44318 Answers: 1 Comments: 2

let f(x)=∫_x ^(+∞) (e^(−t) /t)dt 1)calculate f^′ (x) 2)find a equivalent of f(x) when x→+∞.

$${let}\:{f}\left({x}\right)=\int_{{x}} ^{+\infty} \:\frac{{e}^{−{t}} }{{t}}{dt} \\ $$$$\left.\mathrm{1}\right){calculate}\:{f}^{'} \left({x}\right) \\ $$$$\left.\mathrm{2}\right){find}\:{a}\:{equivalent}\:{of}\:{f}\left({x}\right)\:{when} \\ $$$${x}\rightarrow+\infty. \\ $$

Question Number 44309 Answers: 0 Comments: 2

find the value of I =∫_(−∞) ^(+∞) ((cos(αt))/((x^2 +x +1)^2 ))dx α from R. 2)calculate ∫_(−∞) ^(+∞) (dx/((x^2 +x +1)^2 ))

$${find}\:{the}\:{value}\:{of}\: \\ $$$${I}\:=\int_{−\infty} ^{+\infty} \:\:\:\:\frac{{cos}\left(\alpha{t}\right)}{\left({x}^{\mathrm{2}} \:+{x}\:+\mathrm{1}\right)^{\mathrm{2}} }{dx} \\ $$$$\alpha\:{from}\:{R}. \\ $$$$\left.\mathrm{2}\right){calculate}\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} \:+{x}\:+\mathrm{1}\right)^{\mathrm{2}} } \\ $$

Question Number 44308 Answers: 1 Comments: 2

let I = ∫_0 ^∞ cos^4 t e^(−2t) dt and J=∫_0 ^∞ sin^4 t e^(−2t) dt 1) calculate I +J and I−J 2)find the values of I and J.

$${let}\:{I}\:=\:\int_{\mathrm{0}} ^{\infty} \:{cos}^{\mathrm{4}} {t}\:{e}^{−\mathrm{2}{t}} {dt}\:{and}\:{J}=\int_{\mathrm{0}} ^{\infty} \:{sin}^{\mathrm{4}} {t}\:{e}^{−\mathrm{2}{t}} {dt} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{I}\:+{J}\:{and}\:{I}−{J} \\ $$$$\left.\mathrm{2}\right){find}\:{the}\:{values}\:{of}\:{I}\:{and}\:{J}. \\ $$

Question Number 44307 Answers: 0 Comments: 0

find ∫_0 ^(π/2) cosxln(cosx)dx

$${find}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {cosxln}\left({cosx}\right){dx} \\ $$

Question Number 44306 Answers: 0 Comments: 2

find ∫ (dt/((t+1)(√t) +t(√(t+1)))) 2) calculate ∫_1 ^3 (dt/((t+1)(√t)+t(√(t+1))))

$${find}\:\int\:\:\frac{{dt}}{\left({t}+\mathrm{1}\right)\sqrt{{t}}\:+{t}\sqrt{{t}+\mathrm{1}}} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\int_{\mathrm{1}} ^{\mathrm{3}} \:\:\frac{{dt}}{\left({t}+\mathrm{1}\right)\sqrt{{t}}+{t}\sqrt{{t}+\mathrm{1}}} \\ $$

Question Number 44305 Answers: 0 Comments: 1

let f(a) =∫_0 ^∞ ln(1+(a^2 /x^2 ))dx 1) find a explicit form of f(x) 2)find ∫_0 ^∞ ln(1+(1/x^2 ))dx 3)calculate ∫_0 ^∞ ln(1+(2/x^2 ))dx

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

Question Number 44304 Answers: 1 Comments: 2

find f(a)=∫_0 ^(π/4) (dx/(1+acos^2 x)) a from R.

$${find}\:{f}\left({a}\right)=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\frac{{dx}}{\mathrm{1}+{acos}^{\mathrm{2}} {x}} \\ $$$${a}\:{from}\:{R}. \\ $$

Question Number 44302 Answers: 2 Comments: 1

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

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

Question Number 44317 Answers: 1 Comments: 0

let u_n =∫_0 ^∞ ((t−[t])/(t(t+n)))dt find a equivalent of u_n when n→+∞

$${let}\:{u}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\frac{{t}−\left[{t}\right]}{{t}\left({t}+{n}\right)}{dt} \\ $$$${find}\:{a}\:{equivalent}\:{of}\:{u}_{{n}} \:{when}\:{n}\rightarrow+\infty \\ $$

Question Number 44264 Answers: 1 Comments: 1

∫(1/((x^2 +2x+5)^2 ))dx

$$\int\frac{\mathrm{1}}{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{2x}+\mathrm{5}\right)^{\mathrm{2}} }\mathrm{dx} \\ $$

Question Number 44250 Answers: 1 Comments: 0

Evaliate ∫sin^4 x dx please show working.

$$\mathrm{Evaliate}\:\int\mathrm{sin}^{\mathrm{4}} \mathrm{x}\:\mathrm{dx} \\ $$$$\mathrm{please}\:\mathrm{show}\:\mathrm{working}. \\ $$

Question Number 44232 Answers: 1 Comments: 0

∫_0 ^π e^(sin^2 x) Cos^3 xdx

$$\int_{\mathrm{0}} ^{\pi} \mathrm{e}^{\mathrm{sin}^{\mathrm{2}} \mathrm{x}} \mathrm{Cos}^{\mathrm{3}} \mathrm{xdx} \\ $$

Question Number 44202 Answers: 1 Comments: 4

find f(a) =∫_0 ^∞ (dx/(x^3 +a^3 )) with a>0 2)find g(a)=∫_0 ^∞ (dx/((x^3 +a^3 )^2 )) 3)find the value of ∫_0 ^∞ (dx/((1+x^3 )^2 )) 4)find the value of ∫_0 ^∞ (dx/(8x^3 +1))

$${find}\:\:{f}\left({a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{{x}^{\mathrm{3}} \:+{a}^{\mathrm{3}} }\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right){find}\:{g}\left({a}\right)=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\left({x}^{\mathrm{3}} \:+{a}^{\mathrm{3}} \right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{3}\right){find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{3}} \right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{4}\right){find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\mathrm{8}{x}^{\mathrm{3}} \:+\mathrm{1}} \\ $$

Question Number 44201 Answers: 1 Comments: 4

let f(x)=∫_0 ^1 ((ln(1+xt^2 ))/t^2 )dt with x ∈R 1) find a explicit form of f(x) 2)calculate ∫_0 ^1 ((ln(1+t^2 ))/t^2 )dt 3)calculate ∫_0 ^1 ((ln(1+2t^2 ))/t^2 )dt 4) calculate ∫_0 ^1 ((ln(1−t^2 ))/t^2 )dt

$${let}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{ln}\left(\mathrm{1}+{xt}^{\mathrm{2}} \right)}{{t}^{\mathrm{2}} }{dt}\:\:{with}\:{x}\:\in{R} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{ln}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)}{{t}^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{3}\right){calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{ln}\left(\mathrm{1}+\mathrm{2}{t}^{\mathrm{2}} \right)}{{t}^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{ln}\left(\mathrm{1}−{t}^{\mathrm{2}} \right)}{{t}^{\mathrm{2}} }{dt} \\ $$

Question Number 44179 Answers: 1 Comments: 1

1) find ∫ (dx/((√(x^2 +x+1))+(√(x^2 −x+1)))) 2)calculate ∫_0 ^1 (dx/((√(x^2 +x+1))+(√(x^2 −x +1))))

$$\left.\mathrm{1}\right)\:{find}\:\int\:\:\:\frac{{dx}}{\sqrt{{x}^{\mathrm{2}} \:+{x}+\mathrm{1}}+\sqrt{{x}^{\mathrm{2}} −{x}+\mathrm{1}}} \\ $$$$\left.\mathrm{2}\right){calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\:\frac{{dx}}{\sqrt{{x}^{\mathrm{2}} \:+{x}+\mathrm{1}}+\sqrt{{x}^{\mathrm{2}} −{x}\:+\mathrm{1}}} \\ $$

Question Number 44178 Answers: 0 Comments: 0

find f(x)=∫_0 ^π ((sin^2 t)/((x^2 −2x cost +1)^2 ))dt 2)find the value of ∫_0 ^π ((sin^2 t)/((x^2 −cost +1)^2 ))dt

$${find}\:\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\pi} \:\:\frac{{sin}^{\mathrm{2}} {t}}{\left({x}^{\mathrm{2}} −\mathrm{2}{x}\:{cost}\:+\mathrm{1}\right)^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{2}\right){find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\pi} \:\:\frac{{sin}^{\mathrm{2}} {t}}{\left({x}^{\mathrm{2}} −{cost}\:+\mathrm{1}\right)^{\mathrm{2}} }{dt} \\ $$

Question Number 44176 Answers: 0 Comments: 1

find A_n =∫_0 ^∞ (t^n −[t])e^(−nt) dt and lim_(n→+∞) A_n n integr natural.

$${find}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\infty} \left({t}^{{n}} −\left[{t}\right]\right){e}^{−{nt}} {dt} \\ $$$${and}\:{lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} \\ $$$${n}\:{integr}\:{natural}. \\ $$

Question Number 44175 Answers: 0 Comments: 2

find f(a) =∫_1 ^(+∞) (dx/(ch^2 x +a sh^2 x)) 2) find the value of ∫_1 ^(+∞) (dx/(ch^2 x+2sh^2 x))

$${find}\:\:{f}\left({a}\right)\:=\int_{\mathrm{1}} ^{+\infty} \:\:\frac{{dx}}{{ch}^{\mathrm{2}} {x}\:+{a}\:{sh}^{\mathrm{2}} {x}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{1}} ^{+\infty} \:\:\:\frac{{dx}}{{ch}^{\mathrm{2}} {x}+\mathrm{2}{sh}^{\mathrm{2}} {x}} \\ $$

Question Number 44174 Answers: 0 Comments: 0

find ∫_(−∞) ^(+∞) (dx/((1+x^2 )(1+x e^(iθ) )))

$${find}\:\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\mathrm{1}+{x}\:{e}^{{i}\theta} \right)} \\ $$

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