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IntegrationQuestion and Answers: Page 292

Question Number 36946 Answers: 0 Comments: 2

calculateϕ(λ)= ∫_0 ^π ((cos(t))/(1−2λ cost +λ^2 )) dt

$${calculate}\varphi\left(\lambda\right)=\:\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\:\:\:\frac{{cos}\left({t}\right)}{\mathrm{1}−\mathrm{2}\lambda\:{cost}\:+\lambda^{\mathrm{2}} }\:{dt} \\ $$

Question Number 36945 Answers: 0 Comments: 0

calulate ∫_0 ^(π/4) (dx/(√(tan(x)(1−tanx))))

$${calulate}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\:\frac{{dx}}{\sqrt{{tan}\left({x}\right)\left(\mathrm{1}−{tanx}\right)}} \\ $$

Question Number 36944 Answers: 1 Comments: 1

find ϕ(a) = ∫_a ^(+∞) (dx/((1+x^2 )(√(x^2 −a^2 )))) with a>0

$${find}\:\varphi\left({a}\right)\:=\:\int_{{a}} ^{+\infty} \:\:\:\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\sqrt{{x}^{\mathrm{2}} \:−{a}^{\mathrm{2}} }}\:\:\:{with}\:{a}>\mathrm{0} \\ $$

Question Number 36943 Answers: 0 Comments: 1

find the value of ∫_0 ^1 ((lnx)/((√x)(1−x)^(3/2) ))dx

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{lnx}}{\sqrt{{x}}\left(\mathrm{1}−{x}\right)^{\frac{\mathrm{3}}{\mathrm{2}}} }{dx} \\ $$

Question Number 36942 Answers: 1 Comments: 0

let I = ∫_0 ^(π/2) ((cosx)/(√(1+cosx sinx)))dx and J =∫_0 ^(π/2) ((sinx)/(√(1+cosx sinx)))dx prove that I=J then calculate I and J .

$${let}\:{I}\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{cosx}}{\sqrt{\mathrm{1}+{cosx}\:{sinx}}}{dx}\:{and}\:{J}\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{sinx}}{\sqrt{\mathrm{1}+{cosx}\:{sinx}}}{dx} \\ $$$${prove}\:{that}\:{I}={J}\:\:{then}\:{calculate}\:{I}\:{and}\:{J}\:. \\ $$

Question Number 36941 Answers: 0 Comments: 1

calculate ∫_0 ^((3π)/4) (dt/((1+sin^2 t)^2 ))

$${calculate}\:\:\int_{\mathrm{0}} ^{\frac{\mathrm{3}\pi}{\mathrm{4}}} \:\:\:\:\:\:\frac{{dt}}{\left(\mathrm{1}+{sin}^{\mathrm{2}} {t}\right)^{\mathrm{2}} } \\ $$

Question Number 36940 Answers: 0 Comments: 1

find f(a)= ∫_0 ^a arctan((√(a^2 −x^2 )))dx

$${find}\:{f}\left({a}\right)=\:\int_{\mathrm{0}} ^{{a}} \:{arctan}\left(\sqrt{{a}^{\mathrm{2}} \:−{x}^{\mathrm{2}} }\right){dx} \\ $$

Question Number 36939 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 36938 Answers: 0 Comments: 2

1) find f(a) = ∫_0 ^(π/2) (dt/(1+a cost)) 2) find A(θ) =∫_0 ^(π/2) (dt/(1+sinθ cost))

$$\left.\mathrm{1}\right)\:{find}\:\:\:{f}\left({a}\right)\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{dt}}{\mathrm{1}+{a}\:{cost}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{A}\left(\theta\right)\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\:\frac{{dt}}{\mathrm{1}+{sin}\theta\:{cost}} \\ $$

Question Number 36937 Answers: 0 Comments: 1

calculate ∫_0 ^π ((x dx)/(1+cosx))

$${calculate}\:\:\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{x}\:{dx}}{\mathrm{1}+{cosx}} \\ $$

Question Number 36936 Answers: 0 Comments: 2

calculate I_n = ∫_0 ^π (dx/(1+cos^2 (nx)))

$${calculate}\:{I}_{{n}} \:=\:\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{dx}}{\mathrm{1}+{cos}^{\mathrm{2}} \left({nx}\right)} \\ $$

Question Number 36935 Answers: 0 Comments: 0

find all function f R→R wich verify ∀(x,y)∈ R^2 f(x).f(y) =∫_(x−y) ^(x+y) f(t)dt .

$${find}\:{all}\:{function}\:{f}\:{R}\rightarrow{R}\:\:{wich}\:{verify} \\ $$$$\forall\left({x},{y}\right)\in\:{R}^{\mathrm{2}} \:\:\:{f}\left({x}\right).{f}\left({y}\right)\:=\int_{{x}−{y}} ^{{x}+{y}} \:{f}\left({t}\right){dt}\:. \\ $$

Question Number 36932 Answers: 0 Comments: 0

let f ∈ C^0 ([0,π],R) prove that lim_(n→+∞) ∫_0 ^π f(x) ∣sin(nx)∣dx =(2/π) ∫_0 ^π f(x)dx .

$${let}\:{f}\:\in\:{C}^{\mathrm{0}} \left(\left[\mathrm{0},\pi\right],{R}\right)\:\:{prove}\:{that} \\ $$$${lim}_{{n}\rightarrow+\infty} \:\int_{\mathrm{0}} ^{\pi} {f}\left({x}\right)\:\mid{sin}\left({nx}\right)\mid{dx}\:=\frac{\mathrm{2}}{\pi}\:\int_{\mathrm{0}} ^{\pi} {f}\left({x}\right){dx}\:. \\ $$

Question Number 36931 Answers: 0 Comments: 2

calculate ∫_0 ^(2π) (dt/(x −e^(it) ))

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

Question Number 36930 Answers: 0 Comments: 0

let u_n = (1/(2n+1)) +(1/(2n+3)) +.....+(1/(4n−1)) calculate lim_(n→+∞) u_n .

$${let}\:{u}_{{n}} =\:\frac{\mathrm{1}}{\mathrm{2}{n}+\mathrm{1}}\:+\frac{\mathrm{1}}{\mathrm{2}{n}+\mathrm{3}}\:+.....+\frac{\mathrm{1}}{\mathrm{4}{n}−\mathrm{1}} \\ $$$${calculate}\:{lim}_{{n}\rightarrow+\infty} \:{u}_{{n}} . \\ $$

Question Number 36919 Answers: 0 Comments: 1

calculate f(α)= ∫_(−∞) ^(+∞) (1+αi)^(−x^2 ) dx .

$${calculate}\:{f}\left(\alpha\right)=\:\int_{−\infty} ^{+\infty} \:\left(\mathrm{1}+\alpha{i}\right)^{−{x}^{\mathrm{2}} } {dx}\:. \\ $$

Question Number 36918 Answers: 0 Comments: 0

calculate ∫_0 ^(+∞) (1−i)^(−x^2 ) dx

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

Question Number 36917 Answers: 0 Comments: 1

calculate ∫_0 ^(+∞) (1+i)^(−x^2 ) dx

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

Question Number 36916 Answers: 0 Comments: 1

let z=r e^(iθ) fins f(z) = ∫_(−∞) ^(+∞) z^(−x^2 ) dx

$${let}\:{z}={r}\:{e}^{{i}\theta} \:\:\:\:{fins}\:{f}\left({z}\right)\:=\:\int_{−\infty} ^{+\infty} \:\:{z}^{−{x}^{\mathrm{2}} } {dx} \\ $$

Question Number 36915 Answers: 0 Comments: 1

let z =a+ib find f(z) = ∫_(−∞) ^(+∞) z^(−x^2 ) dx

$${let}\:{z}\:={a}+{ib}\:\:\:{find}\:\:{f}\left({z}\right)\:=\:\int_{−\infty} ^{+\infty} \:{z}^{−{x}^{\mathrm{2}} } {dx} \\ $$

Question Number 36912 Answers: 0 Comments: 1

let ⟨p,q⟩= ∫_(−1) ^1 p(x)q(x)dx with p and q are two polynoms fromR[x] 1)let p(x)=x^n calculate ⟨p,p⟩ 2)let p(x)=1+x+x^2 +....+x^n find ⟨p,p⟩.

$${let}\:\:\langle{p},{q}\rangle=\:\int_{−\mathrm{1}} ^{\mathrm{1}} {p}\left({x}\right){q}\left({x}\right){dx}\:\:{with}\:{p}\:{and}\:{q}\:{are} \\ $$$${two}\:{polynoms}\:{fromR}\left[{x}\right] \\ $$$$\left.\mathrm{1}\right){let}\:{p}\left({x}\right)={x}^{{n}} \:\:\:{calculate}\:\langle{p},{p}\rangle \\ $$$$\left.\mathrm{2}\right){let}\:{p}\left({x}\right)=\mathrm{1}+{x}+{x}^{\mathrm{2}} \:+....+{x}^{{n}} \\ $$$${find}\:\langle{p},{p}\rangle. \\ $$

Question Number 36910 Answers: 0 Comments: 0

1) decompose inside R(x) the fraction F(x)= (1/((1−x^2 )(1−x^3 ))) 2) find ∫ F(x)dx .

$$\left.\mathrm{1}\right)\:{decompose}\:{inside}\:{R}\left({x}\right)\:{the}\:{fraction} \\ $$$${F}\left({x}\right)=\:\:\frac{\mathrm{1}}{\left(\mathrm{1}−{x}^{\mathrm{2}} \right)\left(\mathrm{1}−{x}^{\mathrm{3}} \right)} \\ $$$$\left.\mathrm{2}\right)\:{find}\:\int\:{F}\left({x}\right){dx}\:. \\ $$

Question Number 36892 Answers: 1 Comments: 1

2. ∫[(√((1−x^2 )/(1+x^2 )))]dx=?

$$\mathrm{2}.\:\int\left[\sqrt{\left(\mathrm{1}−{x}^{\mathrm{2}} \right)/\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}\right]{dx}=? \\ $$

Question Number 36818 Answers: 1 Comments: 1

find f(a) = ∫ (dx/(√(1−ax^2 ))) with a from R .

$${find}\:{f}\left({a}\right)\:=\:\int\:\:\:\:\:\frac{{dx}}{\sqrt{\mathrm{1}−{ax}^{\mathrm{2}} }}\:\:{with}\:{a}\:{from}\:{R}\:. \\ $$

Question Number 36811 Answers: 1 Comments: 0

∫ ((sin x)/(cos^2 x. (√(cos 2x)))) dx= ?

$$\int\:\frac{\mathrm{sin}\:{x}}{\mathrm{cos}\:^{\mathrm{2}} {x}.\:\sqrt{\mathrm{cos}\:\mathrm{2}{x}}}\:{dx}=\:? \\ $$

Question Number 36801 Answers: 2 Comments: 0

∫ ((1+x^4 )/((1−x^4 )^(3/2) )) dx = A ∫ A = B Find B ? Assume integration of constant=0.

$$\int\:\frac{\mathrm{1}+{x}^{\mathrm{4}} }{\left(\mathrm{1}−{x}^{\mathrm{4}} \right)^{\frac{\mathrm{3}}{\mathrm{2}}} }\:{dx}\:=\:{A}\: \\ $$$$\int\:\mathrm{A}\:=\:\mathrm{B} \\ $$$$\mathrm{Find}\:\mathrm{B}\:? \\ $$$$\mathrm{Assume}\:\mathrm{integration}\:\mathrm{of}\:\mathrm{constant}=\mathrm{0}. \\ $$

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