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IntegrationQuestion and Answers: Page 296

Question Number 32361 Answers: 0 Comments: 1

let give a>0 find ∫_0 ^∞ (e^(−x) /(√(x+a))) dx.

$${let}\:{give}\:{a}>\mathrm{0}\:{find}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{e}^{−{x}} }{\sqrt{{x}+{a}}}\:{dx}. \\ $$

Question Number 32360 Answers: 0 Comments: 1

find the value of ∫_0 ^∞ (dx/((2x+1)(2x+3))) .

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dx}}{\left(\mathrm{2}{x}+\mathrm{1}\right)\left(\mathrm{2}{x}+\mathrm{3}\right)}\:. \\ $$

Question Number 32359 Answers: 0 Comments: 1

find ∫_0 ^∞ (dx/((1+x^2 )(1+x^4 ))) .

$${find}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\mathrm{1}+{x}^{\mathrm{4}} \right)}\:. \\ $$

Question Number 32354 Answers: 0 Comments: 1

calculate ∫_0 ^(π/4) (dt/((1+sin^2 t)^2 )) .

$${calculate}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\:\:\:\:\:\frac{{dt}}{\left(\mathrm{1}+{sin}^{\mathrm{2}} {t}\right)^{\mathrm{2}} }\:. \\ $$

Question Number 32353 Answers: 1 Comments: 0

calculate ∫_0 ^(π/4) cos(x)ln(cos(x))dx .

$${calculate}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:{cos}\left({x}\right){ln}\left({cos}\left({x}\right)\right){dx}\:. \\ $$

Question Number 32352 Answers: 1 Comments: 2

find the value of ∫_0 ^1 arctan((√(1−x^2 )))dx

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{arctan}\left(\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }\right){dx} \\ $$

Question Number 32351 Answers: 1 Comments: 0

calculate ∫_0 ^(π/2) (dt/(1+cosθ sint)) .

$${calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\:\:\:\frac{{dt}}{\mathrm{1}+{cos}\theta\:{sint}}\:.\: \\ $$

Question Number 32350 Answers: 0 Comments: 0

calculate ∫_0 ^1 ^3 (√(x^2 (1−x))) dx

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

Question Number 32349 Answers: 0 Comments: 1

find the value of ∫_0 ^π ((xdx)/(1+sinx)) .

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{xdx}}{\mathrm{1}+{sinx}}\:. \\ $$

Question Number 32343 Answers: 1 Comments: 1

calculate ∫_0 ^(2π) (dt/(x−e^(it) )) .

$${calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\:\frac{{dt}}{{x}−{e}^{{it}} }\:\:. \\ $$

Question Number 32342 Answers: 0 Comments: 0

find the value of ∫∫_D ((dxdy)/((4x^2 +y^2 +1)^2 )) D={(x,y)∈ R^2 / x^2 +y^2 ≤1 and y ≤2x } .

$${find}\:{the}\:{value}\:{of}\:\int\int_{{D}} \:\:\:\frac{{dxdy}}{\left(\mathrm{4}{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} } \\ $$$${D}=\left\{\left({x},{y}\right)\in\:{R}^{\mathrm{2}} \:/\:{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \:\leqslant\mathrm{1}\:{and}\:{y}\:\leqslant\mathrm{2}{x}\:\right\}\:. \\ $$

Question Number 32341 Answers: 0 Comments: 1

let give λ from R and λ^2 ≠1 and I_n (λ) = ∫_0 ^π ((cos(nt))/(1−2λcost +λ^2 ))dt .calculate I_n (λ).

$${let}\:{give}\:\lambda\:{from}\:{R}\:{and}\:\lambda^{\mathrm{2}} \neq\mathrm{1}\:{and} \\ $$$${I}_{{n}} \left(\lambda\right)\:=\:\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{cos}\left({nt}\right)}{\mathrm{1}−\mathrm{2}\lambda{cost}\:+\lambda^{\mathrm{2}} }{dt}\:\:.{calculate}\:{I}_{{n}} \left(\lambda\right). \\ $$

Question Number 32340 Answers: 0 Comments: 0

find the value of ∫_0 ^∞ ((sin^3 t)/t^2 ) dt .

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{sin}^{\mathrm{3}} {t}}{{t}^{\mathrm{2}} }\:{dt}\:. \\ $$

Question Number 32339 Answers: 0 Comments: 1

calculate ∫_0 ^(+∞) ((th(3x) −th(2x))/x) dx .

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

Question Number 32338 Answers: 0 Comments: 0

find the value of ∫_0 ^1 ((ln(t))/((1+t)(√(1−t^2 )))) dt.

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

Question Number 32337 Answers: 0 Comments: 0

1)calculate ∫_a ^(+∞) (dx/((1+x^2 )(√(x^2 −a^2 )))) with a>0 2) find the value of ∫_2 ^(+∞) (dx/((1+x^2 )(√(x^2 −4)))) .

$$\left.\mathrm{1}\right){calculate}\:\int_{{a}} ^{+\infty} \:\:\:\:\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\sqrt{{x}^{\mathrm{2}} \:−{a}^{\mathrm{2}} }}\:\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\:\int_{\mathrm{2}} ^{+\infty} \:\:\:\:\:\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\sqrt{{x}^{\mathrm{2}} \:−\mathrm{4}}}\:\:. \\ $$

Question Number 32323 Answers: 1 Comments: 0

Given f(x) = (3/(16))(∫_0 ^1 f(x)dx)x^2 − (9/(10))(∫_0 ^2 f(x)dx)x + 2(∫_0 ^3 f(x)dx) + 4 Find f(x)

$$\mathrm{Given} \\ $$$${f}\left({x}\right)\:=\:\frac{\mathrm{3}}{\mathrm{16}}\left(\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){dx}\right){x}^{\mathrm{2}} \:−\:\frac{\mathrm{9}}{\mathrm{10}}\left(\int_{\mathrm{0}} ^{\mathrm{2}} {f}\left({x}\right){dx}\right){x}\:+\:\mathrm{2}\left(\int_{\mathrm{0}} ^{\mathrm{3}} {f}\left({x}\right){dx}\right)\:+\:\mathrm{4} \\ $$$$\mathrm{Find}\:{f}\left({x}\right) \\ $$

Question Number 32304 Answers: 0 Comments: 0

find lim_(x→+∞) e^(−x^2 ) ∫_0 ^x e^t^2 dt .

$${find}\:{lim}_{{x}\rightarrow+\infty} \:\:{e}^{−{x}^{\mathrm{2}} } \:\int_{\mathrm{0}} ^{{x}} \:\:{e}^{{t}^{\mathrm{2}} } {dt}\:\:. \\ $$

Question Number 32302 Answers: 1 Comments: 0

calculate ∫_1 ^2 (dx/(x +x(√x))) .

$${calculate}\:\:\int_{\mathrm{1}} ^{\mathrm{2}} \:\:\:\:\frac{{dx}}{{x}\:+{x}\sqrt{{x}}}\:. \\ $$

Question Number 32301 Answers: 0 Comments: 1

calculate ∫_1 ^e ln(1+(√x))dx .

$${calculate}\:\int_{\mathrm{1}} ^{{e}} \:{ln}\left(\mathrm{1}+\sqrt{{x}}\right){dx}\:. \\ $$

Question Number 32269 Answers: 1 Comments: 0

find ∫ (x^3 /(√(1+x^2 ))) dx

$${find}\:\int\:\:\frac{{x}^{\mathrm{3}} }{\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}\:{dx} \\ $$

Question Number 32258 Answers: 2 Comments: 0

find ∫ (1/(2−x^2 )) dx

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

Question Number 32206 Answers: 0 Comments: 0

Find Σ_(k=1) ^∞ (∫_(k−1) ^k x^(−x) dx) .

$$\mathrm{Find}\:\underset{\mathrm{k}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\underset{\mathrm{k}−\mathrm{1}} {\overset{\mathrm{k}} {\int}}\mathrm{x}^{−\mathrm{x}} \:\mathrm{dx}\right)\:. \\ $$$$ \\ $$

Question Number 32139 Answers: 0 Comments: 4

Find the ∫ ((x+1)/(x^2 +x+1))dx

$${F}\boldsymbol{{ind}}\:\boldsymbol{{the}} \\ $$$$\int\:\frac{{x}+\mathrm{1}}{{x}^{\mathrm{2}} +{x}+\mathrm{1}}{dx} \\ $$

Question Number 32045 Answers: 0 Comments: 0

find lim_(n→∞) ∫_0 ^∞ e^(−t) sin^n t dt .

$${find}\:{lim}_{{n}\rightarrow\infty} \:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{t}} \:{sin}^{{n}} {t}\:{dt}\:\:. \\ $$

Question Number 32044 Answers: 0 Comments: 1

fimd lim_(x→0) (1/x^3 ) ∫_0 ^x t^2 ln(1+sint) dt .

$${fimd}\:\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\:\:\frac{\mathrm{1}}{{x}^{\mathrm{3}} }\:\int_{\mathrm{0}} ^{{x}} \:{t}^{\mathrm{2}} \:{ln}\left(\mathrm{1}+{sint}\right)\:{dt}\:. \\ $$

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