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AllQuestion and Answers: Page 1775

Question Number 31072 Answers: 0 Comments: 0

find ∫_0 ^∞ (dx/(e^x (√(sh(2x))))) dx.

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

Question Number 31071 Answers: 1 Comments: 3

find ∫_0 ^π (dx/(1+sin^2 x)) .

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

Question Number 31070 Answers: 0 Comments: 1

calculate ∫_0 ^π (dx/(1+2cosx)) .

$${calculate}\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\:\frac{{dx}}{\mathrm{1}+\mathrm{2}{cosx}}\:. \\ $$

Question Number 31069 Answers: 1 Comments: 1

clculate ∫_0 ^1 x(√(x^2 −2x+2)) dx

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

Question Number 31068 Answers: 0 Comments: 0

find I_n =∫_(−(π/2)) ^(π/2) e^(−ax) cos^(2n) xdx .

$${find}\:\:{I}_{{n}} =\int_{−\frac{\pi}{\mathrm{2}}} ^{\frac{\pi}{\mathrm{2}}} \:{e}^{−{ax}} \:{cos}^{\mathrm{2}{n}} {xdx}\:\:. \\ $$

Question Number 31067 Answers: 0 Comments: 0

find A_n =∫_0 ^∞ x^(2n) e^(−ax^2 ) dx.

$${find}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:{x}^{\mathrm{2}{n}} \:{e}^{−{ax}^{\mathrm{2}} } {dx}. \\ $$

Question Number 31066 Answers: 0 Comments: 0

find I_n =∫_0 ^(π/2) cos^(2n+1) xdx.

$${find}\:\:{I}_{{n}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:{cos}^{\mathrm{2}{n}+\mathrm{1}} {xdx}. \\ $$

Question Number 31065 Answers: 0 Comments: 0

find ∫_0 ^π ((xsinx)/((1−acosx)^2 )) dx with ∣a∣<1.

$${find}\:\:\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{xsinx}}{\left(\mathrm{1}−{acosx}\right)^{\mathrm{2}} }\:{dx}\:{with}\:\:\mid{a}\mid<\mathrm{1}. \\ $$

Question Number 31063 Answers: 0 Comments: 0

find f(t)= ∫_0 ^1 ln(1+tx^2 )dxfor t>−1

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

Question Number 31062 Answers: 0 Comments: 0

find ∫_0 ^(π/2) e^x sinx cos^2 xdx.

$${find}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:{e}^{{x}} \:{sinx}\:{cos}^{\mathrm{2}} {xdx}. \\ $$

Question Number 31061 Answers: 0 Comments: 0

find ∫_0 ^(π/2) (sinθ −cosθ)ln(sinθ+cosθ)dθ.

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

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 31047 Answers: 0 Comments: 0

let Δ={(x,y)∈N^2 /x+y=n , n∈N} find cardΔ 2) let A= {(x,y)∈N^2 / x+2y=n} find card A.

$${let}\:\Delta=\left\{\left({x},{y}\right)\in{N}^{\mathrm{2}} \:/{x}+{y}={n}\:,\:{n}\in{N}\right\}\:{find}\:{card}\Delta \\ $$$$\left.\mathrm{2}\right)\:{let}\:{A}=\:\left\{\left({x},{y}\right)\in{N}^{\mathrm{2}} /\:{x}+\mathrm{2}{y}={n}\right\}\:{find}\:{card}\:{A}. \\ $$

Question Number 31046 Answers: 0 Comments: 0

prove that C_n ^o C_n ^p +C_n ^1 C_(n−1) ^(p−1) +...C_n ^p C_(n−p) ^0 =2^p C_n ^p .

$${prove}\:{that}\:{C}_{{n}} ^{{o}} \:{C}_{{n}} ^{{p}} \:+{C}_{{n}} ^{\mathrm{1}} \:{C}_{{n}−\mathrm{1}} ^{{p}−\mathrm{1}} \:+...{C}_{{n}} ^{{p}} \:{C}_{{n}−{p}} ^{\mathrm{0}} =\mathrm{2}^{{p}} \:{C}_{{n}} ^{{p}} \:\:\:. \\ $$

Question Number 31045 Answers: 0 Comments: 0

simlify A_n =(C_n ^1 )^2 +2(C_n ^2 )^2 +n (C_n ^n )^2 .

$${simlify}\:{A}_{{n}} =\left({C}_{{n}} ^{\mathrm{1}} \right)^{\mathrm{2}} \:+\mathrm{2}\left({C}_{{n}} ^{\mathrm{2}} \right)^{\mathrm{2}} \:\:+{n}\:\left({C}_{{n}} ^{{n}} \right)^{\mathrm{2}} . \\ $$

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