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IntegrationQuestion and Answers: Page 297

Question Number 36167 Answers: 0 Comments: 2

let give I = ∫_0 ^∞ (dx/((x^2 +i)^2 )) 1) extract Re(I) and Im(I) 2) find the value of I 3) calculate Re(I) and Im(I) .

$${let}\:{give}\:\:{I}\:\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} \:+{i}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{1}\right)\:{extract}\:{Re}\left({I}\right)\:{and}\:{Im}\left({I}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:{I} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:{Re}\left({I}\right)\:{and}\:{Im}\left({I}\right)\:. \\ $$

Question Number 36057 Answers: 2 Comments: 1

find the value of ∫_0 ^(π/4) ((cosx)/(sinx +tanx))dx

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\:\:\frac{{cosx}}{{sinx}\:+{tanx}}{dx}\: \\ $$

Question Number 36056 Answers: 1 Comments: 2

calculate ∫_(−∞) ^(+∞) ((2x)/((x^2 +mx +1)^2 ))dx with ∣m∣<2

$${calculate}\:\:\int_{−\infty} ^{+\infty} \:\:\:\frac{\mathrm{2}{x}}{\left({x}^{\mathrm{2}} \:+{mx}\:+\mathrm{1}\right)^{\mathrm{2}} }{dx}\:{with}\:\mid{m}\mid<\mathrm{2} \\ $$

Question Number 36031 Answers: 0 Comments: 0

Q. Evaluate: ∫_(∫xyzdxdydz) ^(∫zyxdzdydx) ∫_((d/dx)(x^(sin x) )) ^((d/dx)(x^(cos x) )) ∫_(lim_(x→0) ((−x^2 +2)/x)) ^(lim_(x→0) ((x^2 −2)/x)) ∫_0 ^∞ w^(1−x) x^(1−y) y^(1−z) z^(1−w) dwdxdydz

$$\mathrm{Q}.\:\mathrm{Evaluate}:\:\:\:\int_{\int\mathrm{xyzdxdydz}} ^{\int\mathrm{zyxdzdydx}} \int_{\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{x}^{\mathrm{sin}\:\mathrm{x}} \right)} ^{\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{x}^{\mathrm{cos}\:\mathrm{x}} \right)} \int_{\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{−\mathrm{x}^{\mathrm{2}} +\mathrm{2}}{\mathrm{x}}} ^{\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{x}^{\mathrm{2}} −\mathrm{2}}{\mathrm{x}}} \int_{\mathrm{0}} ^{\infty} \mathrm{w}^{\mathrm{1}−\mathrm{x}} \mathrm{x}^{\mathrm{1}−\mathrm{y}} \mathrm{y}^{\mathrm{1}−\mathrm{z}} \mathrm{z}^{\mathrm{1}−\mathrm{w}} \mathrm{dwdxdydz} \\ $$

Question Number 36009 Answers: 0 Comments: 1

calculate ∫_(−∞) ^(+∞) ((xdx)/((2x+1+i)^3 )) with i^2 =−1 .

$${calculate}\:\int_{−\infty} ^{+\infty} \:\:\:\:\:\frac{{xdx}}{\left(\mathrm{2}{x}+\mathrm{1}+{i}\right)^{\mathrm{3}} }\:\:{with}\:{i}^{\mathrm{2}} \:=−\mathrm{1}\:. \\ $$

Question Number 35990 Answers: 0 Comments: 2

calculate ∫_2 ^5 ((xdx)/(2x+1 +(√(x−1))))

$${calculate}\:\int_{\mathrm{2}} ^{\mathrm{5}} \:\:\:\:\frac{{xdx}}{\mathrm{2}{x}+\mathrm{1}\:+\sqrt{{x}−\mathrm{1}}} \\ $$

Question Number 35983 Answers: 0 Comments: 1

calculate ∫_0 ^∞ ((2(√t) +1)/(t^5 +3))dt .

$${calculate}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{\mathrm{2}\sqrt{{t}}\:+\mathrm{1}}{{t}^{\mathrm{5}} \:\:\:+\mathrm{3}}{dt}\:\:. \\ $$

Question Number 35982 Answers: 0 Comments: 0

let f(t) =∫_0 ^∞ e^(−arctsn( 1+tx^2 )) dx with t from R 1) calculate f^′ (t) 2) find a simple form of f(t) .

$${let}\:{f}\left({t}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:{e}^{−{arctsn}\left(\:\mathrm{1}+{tx}^{\mathrm{2}} \right)} {dx}\:\:{with}\:{t}\:{from}\:{R} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{'} \left({t}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{a}\:{simple}\:{form}\:{of}\:{f}\left({t}\right)\:. \\ $$

Question Number 35949 Answers: 1 Comments: 1

∫_0 ^( α) ((tan θ)/(√(a^2 cos^2 θ−b^2 sin^2 θ))) dθ = ?

$$\int_{\mathrm{0}} ^{\:\:\alpha} \frac{\mathrm{tan}\:\theta}{\sqrt{{a}^{\mathrm{2}} \mathrm{cos}\:^{\mathrm{2}} \theta−{b}^{\mathrm{2}} \mathrm{sin}\:^{\mathrm{2}} \theta}}\:{d}\theta\:=\:? \\ $$

Question Number 35920 Answers: 1 Comments: 1

∫ ((e^(2x) +1)/(2e^x −1)) dx = ?

$$\int\:\frac{{e}^{\mathrm{2}{x}} +\mathrm{1}}{\mathrm{2}{e}^{{x}} −\mathrm{1}}\:{dx}\:=\:? \\ $$

Question Number 35909 Answers: 1 Comments: 4

∫((7x−6)/((x^2 +25)(√((x−3)^2 +4)))) dx = ?

$$\int\frac{\mathrm{7}{x}−\mathrm{6}}{\left({x}^{\mathrm{2}} +\mathrm{25}\right)\sqrt{\left({x}−\mathrm{3}\right)^{\mathrm{2}} +\mathrm{4}}}\:{dx}\:=\:? \\ $$

Question Number 35832 Answers: 1 Comments: 3

find the value of f(λ) = ∫_(−a) ^a (dx/((λ +_ x^2 )^(3/2) )) λ∈R .

$${find}\:{the}\:{value}\:{of}\:\:{f}\left(\lambda\right)\:=\:\int_{−{a}} ^{{a}} \:\:\:\frac{{dx}}{\left(\lambda\:+_{} {x}^{\mathrm{2}} \right)^{\frac{\mathrm{3}}{\mathrm{2}}} } \\ $$$$\lambda\in{R}\:. \\ $$

Question Number 35821 Answers: 0 Comments: 4

let f(t) = ∫_0 ^∞ ((e^(−ax) −e^(−bx) )/x^2 ) e^(−tx^2 ) dx with t>0 1) calculate f^′ (t) 2)find a simple form of f(t) 3) find the value of ∫_0 ^∞ ((e^(−2x) −e^(−x) )/x^2 ) e^(−3x^2 ) dx

$${let}\:{f}\left({t}\right)\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−{ax}} \:−{e}^{−{bx}} }{{x}^{\mathrm{2}} }\:{e}^{−{tx}^{\mathrm{2}} } {dx}\:\:\:{with}\:{t}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{'} \left({t}\right) \\ $$$$\left.\mathrm{2}\right){find}\:{a}\:{simple}\:{form}\:{of}\:{f}\left({t}\right) \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−\mathrm{2}{x}} \:\:−{e}^{−{x}} }{{x}^{\mathrm{2}} }\:{e}^{−\mathrm{3}{x}^{\mathrm{2}} } {dx} \\ $$

Question Number 35732 Answers: 1 Comments: 1