Question and Answers Forum

IntegrationQuestion and Answers: Page 290

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.

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

Question Number 36799 Answers: 1 Comments: 1

find ∫_0 ^∞ e^t ln(1+e^(−2t) )dt .

$${find}\:\int_{\mathrm{0}} ^{\infty} \:\:\:{e}^{{t}} {ln}\left(\mathrm{1}+{e}^{−\mathrm{2}{t}} \right){dt}\:. \\ $$

Question Number 36762 Answers: 1 Comments: 2

find A_n = ∫_0 ^(π/4) (cosx +sinx)^n dx.

$${find}\:{A}_{{n}} \:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\left({cosx}\:+{sinx}\right)^{{n}} \:{dx}. \\ $$

Question Number 36755 Answers: 1 Comments: 4

let f(a) = ∫_0 ^1 e^t ln(1+ e^(−at) )dt with a≥0 1) find f(a) 2) calculate f^′ (a) 3) find the value of ∫_0 ^1 e^t ln(1+e^(−3t) )dt .

$${let}\:{f}\left({a}\right)\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:{e}^{{t}} {ln}\left(\mathrm{1}+\:{e}^{−{at}} \right){dt}\:\:{with}\:{a}\geqslant\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{f}\left({a}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{f}^{'} \left({a}\right) \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{e}^{{t}} {ln}\left(\mathrm{1}+{e}^{−\mathrm{3}{t}} \right){dt}\:. \\ $$

Question Number 36754 Answers: 1 Comments: 1

calculate ∫_1 ^(+∞) (dx/(x^2 (√(4+x^2 )))) .

$${calculate}\:\:\:\int_{\mathrm{1}} ^{+\infty} \:\:\:\:\frac{{dx}}{{x}^{\mathrm{2}} \sqrt{\mathrm{4}+{x}^{\mathrm{2}} }}\:. \\ $$

Question Number 36753 Answers: 1 Comments: 2

find I_n = ∫_0 ^1 x^n arctan(x)dx .

$${find}\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:{x}^{{n}} \:{arctan}\left({x}\right){dx}\:. \\ $$

Question Number 36752 Answers: 1 Comments: 4

find ∫ (dx/(arcsinx(√(1−x^2 )))) .

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

Question Number 36747 Answers: 0 Comments: 1

let f(x)= Σ_(n=1) ^∞ ((sin(nx))/n) x^n 1) prove that f is C^1 on ]−1,1[ 2)calculate f^′ (x) and prove that f(x)=arctan( ((xsinx)/(1−x cosx)))

$${let}\:{f}\left({x}\right)=\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\frac{{sin}\left({nx}\right)}{{n}}\:{x}^{{n}} \\ $$$$\left.\mathrm{1}\left.\right)\:{prove}\:{that}\:{f}\:{is}\:{C}^{\mathrm{1}} \:{on}\:\right]−\mathrm{1},\mathrm{1}\left[\right. \\ $$$$\left.\mathrm{2}\right){calculate}\:{f}^{'} \left({x}\right)\:{and}\:{prove}\:{that} \\ $$$${f}\left({x}\right)={arctan}\left(\:\frac{{xsinx}}{\mathrm{1}−{x}\:{cosx}}\right) \\ $$

Question Number 36738 Answers: 2 Comments: 3

(1) ∫(dα/((1+sin 2α)^2 ))= (2) ∫(dβ/((1+cos 2β)^2 ))= (3) ∫(dγ/((1+sin 2γ)(1+cos 2γ)))=

$$\left(\mathrm{1}\right)\:\:\:\:\:\int\frac{{d}\alpha}{\left(\mathrm{1}+\mathrm{sin}\:\mathrm{2}\alpha\right)^{\mathrm{2}} }= \\ $$$$\left(\mathrm{2}\right)\:\:\:\:\:\int\frac{{d}\beta}{\left(\mathrm{1}+\mathrm{cos}\:\mathrm{2}\beta\right)^{\mathrm{2}} }= \\ $$$$\left(\mathrm{3}\right)\:\:\:\:\:\int\frac{{d}\gamma}{\left(\mathrm{1}+\mathrm{sin}\:\mathrm{2}\gamma\right)\left(\mathrm{1}+\mathrm{cos}\:\mathrm{2}\gamma\right)}= \\ $$

Pg 285 Pg 286 Pg 287 Pg 288 Pg 289 Pg 290 Pg 291 Pg 292 Pg 293 Pg 294

Contact: info@tinkutara.com