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IntegrationQuestion and Answers: Page 312

Question Number 31060 Answers: 0 Comments: 0

calculate by recurrence ∫_0 ^∞ ((lnx)/((1+x)^n ))dx with n≥2 .

$${calculate}\:{by}\:{recurrence}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{lnx}}{\left(\mathrm{1}+{x}\right)^{{n}} }{dx}\:{with}\:{n}\geqslant\mathrm{2}\:. \\ $$

Question Number 31059 Answers: 0 Comments: 0

find ∫_0 ^(π/2) cos(2θ)ln(tanθ)dθ.

$${find}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:{cos}\left(\mathrm{2}\theta\right){ln}\left({tan}\theta\right){d}\theta. \\ $$

Question Number 31058 Answers: 0 Comments: 0

find ∫_0 ^∞ ((x arctanx)/((1+x^2 )^2 ))dx

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

Question Number 31057 Answers: 0 Comments: 0

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

$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\:−\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}{{x}^{\mathrm{2}} }\:{dx}.\: \\ $$

Question Number 31056 Answers: 0 Comments: 1

find ∫_1 ^(+∞) (dx/(x^2 −2xcosα +1)) with 0<α<π .

$${find}\:\:\int_{\mathrm{1}} ^{+\infty} \:\:\:\:\:\:\:\frac{{dx}}{{x}^{\mathrm{2}} \:−\mathrm{2}{xcos}\alpha\:+\mathrm{1}}\:\:{with}\:\mathrm{0}<\alpha<\pi\:. \\ $$

Question Number 31055 Answers: 0 Comments: 1

find ∫_(−∞) ^(+∞) (dx/((x^2 −x+1)(x^2 −2x+4))) .

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

Question Number 31054 Answers: 0 Comments: 0

find ∫_0 ^1 (dx/(x^4 +1)) .

$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{dx}}{{x}^{\mathrm{4}} \:+\mathrm{1}}\:. \\ $$

Question Number 31053 Answers: 0 Comments: 1

let λ ∈R and a>0 find ∫_0 ^∞ e^(−ax) cos(λx)dx .

$${let}\:\lambda\:\in{R}\:{and}\:{a}>\mathrm{0}\:\:{find}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{ax}} {cos}\left(\lambda{x}\right){dx}\:. \\ $$

Question Number 31052 Answers: 0 Comments: 0

let give 0<a<b find ∫_a ^b ((lnx)/x)dx .

$${let}\:{give}\:\mathrm{0}<{a}<{b}\:\:{find}\:\int_{{a}} ^{{b}} \:\:\frac{{lnx}}{{x}}{dx}\:. \\ $$

Question Number 31051 Answers: 0 Comments: 0

study the convergence of ∫_0 ^∞ ((e^(−ax) −e^(−bx) )/(1− e^(−x) )) dx.

$${study}\:{the}\:{convergence}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{e}^{−{ax}} \:−{e}^{−{bx}} }{\mathrm{1}−\:{e}^{−{x}} }\:{dx}. \\ $$

Question Number 31049 Answers: 0 Comments: 0

study the convergence of ∫_0 ^∞ x^(−x) dx .

$${study}\:{the}\:{convergence}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:{x}^{−{x}} {dx}\:. \\ $$

Question Number 31048 Answers: 0 Comments: 0

study the convergence of ∫_1 ^(+∞) (((π/2) −arctanx)/x)dx

$${study}\:{the}\:{convergence}\:{of}\:\int_{\mathrm{1}} ^{+\infty} \:\:\frac{\frac{\pi}{\mathrm{2}}\:−{arctanx}}{{x}}{dx} \\ $$

Question Number 31145 Answers: 1 Comments: 0

Given ∫_0 ^1 f(x) dx = (((2018)),(( 0)) ) + (1/2) (((2018)),(( 1)) ) + (1/3) (((2018)),(( 2)) ) + ... + (1/(2019)) (((2018)),((2018)) ) ∫_0 ^1 g(x) dx = (((2018)),(( 0)) ) − (1/2) (((2018)),(( 1)) ) + (1/3) (((2018)),(( 2)) ) − ... + (1/(2019)) (((2018)),((2018)) ) h(x) is an odd function Then what is the value of ∫_(−3) ^( 3) f(x).g(x).h(x) dx ?

$$\mathrm{Given} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \:{f}\left({x}\right)\:{dx}\:=\:\begin{pmatrix}{\mathrm{2018}}\\{\:\:\:\:\mathrm{0}}\end{pmatrix}\:+\:\frac{\mathrm{1}}{\mathrm{2}}\begin{pmatrix}{\mathrm{2018}}\\{\:\:\:\:\mathrm{1}}\end{pmatrix}\:+\:\frac{\mathrm{1}}{\mathrm{3}}\begin{pmatrix}{\mathrm{2018}}\\{\:\:\:\:\mathrm{2}}\end{pmatrix}\:+\:...\:+\:\frac{\mathrm{1}}{\mathrm{2019}}\begin{pmatrix}{\mathrm{2018}}\\{\mathrm{2018}}\end{pmatrix} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \:{g}\left({x}\right)\:{dx}\:=\:\begin{pmatrix}{\mathrm{2018}}\\{\:\:\:\:\mathrm{0}}\end{pmatrix}\:−\:\frac{\mathrm{1}}{\mathrm{2}}\begin{pmatrix}{\mathrm{2018}}\\{\:\:\:\:\mathrm{1}}\end{pmatrix}\:+\:\frac{\mathrm{1}}{\mathrm{3}}\begin{pmatrix}{\mathrm{2018}}\\{\:\:\:\:\mathrm{2}}\end{pmatrix}\:−\:...\:+\:\frac{\mathrm{1}}{\mathrm{2019}}\begin{pmatrix}{\mathrm{2018}}\\{\mathrm{2018}}\end{pmatrix} \\ $$$${h}\left({x}\right)\:\mathrm{is}\:\mathrm{an}\:\mathrm{odd}\:\mathrm{function} \\ $$$$\mathrm{Then}\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\int_{−\mathrm{3}} ^{\:\mathrm{3}} \:{f}\left({x}\right).{g}\left({x}\right).{h}\left({x}\right)\:{dx}\:? \\ $$

Question Number 31141 Answers: 1 Comments: 0

using the limit defination find the area of f(x)= cos(x) [0,π/2]

$$\boldsymbol{{using}}\:\boldsymbol{{the}}\:\boldsymbol{{limit}}\:\boldsymbol{{defination}} \\ $$$$\boldsymbol{{find}}\:\boldsymbol{{the}}\:\boldsymbol{{area}}\:\boldsymbol{{of}} \\ $$$$\boldsymbol{{f}}\left(\boldsymbol{{x}}\right)=\:\boldsymbol{{cos}}\left(\boldsymbol{{x}}\right)\:\:\left[\mathrm{0},\pi/\mathrm{2}\right] \\ $$

Question Number 30936 Answers: 0 Comments: 0

find ∫^a _0 ((sinx)/(√(1+x^2 )))dx .

$${find}\:\:\:\underset{\mathrm{0}} {\int}^{{a}} \:\:\frac{{sinx}}{\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}{dx}\:. \\ $$

Question Number 30858 Answers: 2 Comments: 0

∫_0 ^1 x∣x−4∣dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{x}\mid\mathrm{x}−\mathrm{4}\mid\mathrm{dx} \\ $$

Question Number 30855 Answers: 1 Comments: 0

∫((cosec^2 (x))/(√(cosecx+cotx)))dx

$$\int\frac{\mathrm{cosec}^{\mathrm{2}} \left(\mathrm{x}\right)}{\sqrt{\mathrm{cosecx}+\mathrm{cotx}}}\mathrm{dx} \\ $$

Question Number 30798 Answers: 0 Comments: 1

find ∫_0 ^1 ((ln(1−x^2 ))/x^2 )dx

$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{ln}\left(\mathrm{1}−{x}^{\mathrm{2}} \right)}{{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 30796 Answers: 0 Comments: 0

find ∫_0 ^∞ e^(−(x^2 +(1/x^2 ))) dx.

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

Question Number 30777 Answers: 0 Comments: 0

find interms of n A_n = ∫_0 ^∞ ((ln(x))/((1+x^ )^n )) dx with n from N and n≥3 .

$${find}\:{interms}\:{of}\:{n}\:\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{\infty} \:\frac{{ln}\left({x}\right)}{\left(\mathrm{1}+{x}^{} \right)^{{n}} }\:{dx}\:{with}\:{n}\:{from} \\ $$$${N}\:{and}\:{n}\geqslant\mathrm{3}\:. \\ $$

Question Number 30776 Answers: 1 Comments: 1

find ∫_0 ^1 ((xdx)/((1+x^2 )(√(1−x^4 )))) .

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

Question Number 30775 Answers: 1 Comments: 1

letα ∈]0,π[ calculate ∫_0 ^(π/2) (dx/(2(cosα +chx))) .

$$\left.{let}\alpha\:\in\right]\mathrm{0},\pi\left[\:\:{calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{dx}}{\mathrm{2}\left({cos}\alpha\:+{chx}\right)}\:.\right. \\ $$

Question Number 30774 Answers: 1 Comments: 1

find f(t)=∫_0 ^1 ln(1+tx^2 )dx with t>0

$${find}\:{f}\left({t}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} \:{ln}\left(\mathrm{1}+{tx}^{\mathrm{2}} \right){dx}\:\:{with}\:{t}>\mathrm{0} \\ $$

Question Number 30773 Answers: 0 Comments: 1

let a>0 calculate ∫_0 ^∞ (dx/((x^2 +a^2 )^2 )) 2) calculate ∫_(−∞) ^(+∞) (x^2 /((x^2 +a^2 )^2 ))dx.

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

Question Number 30769 Answers: 0 Comments: 1

find the value of I= ∫_0 ^1 (dx/((x+1)^2 (√(x^2 +2x +2)))) .

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

Question Number 30767 Answers: 0 Comments: 1

calculate A_n = ∫_0 ^1 ((1−(√x))/(1−^n (√x)))dx.

$${calculate}\:\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{\mathrm{1}−\sqrt{{x}}}{\mathrm{1}−^{{n}} \sqrt{{x}}}{dx}. \\ $$

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