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Question Number 65805 Answers: 2 Comments: 1

Prove that I_n =∫_0 ^(π/2) (dt/(1+(tant)^n )) does not depend of the term n deduces that ∫_0 ^∞ (dx/((x^(2035) +1)(x^2 +1)))=(π/4)

$$\:\:{Prove}\:{that}\:\:{I}_{{n}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{dt}}{\mathrm{1}+\left({tant}\right)^{{n}} }\:\:{does}\:{not}\:{depend}\:{of}\:{the}\:{term}\:{n} \\ $$$${deduces}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\left({x}^{\mathrm{2035}} +\mathrm{1}\right)\left({x}^{\mathrm{2}} +\mathrm{1}\right)}=\frac{\pi}{\mathrm{4}} \\ $$

Question Number 65797 Answers: 1 Comments: 0

find the constant a,b and c so that the direction derivative of Φ=axy^2 +byz+cz^2 x^3 at (1,2,−1) has a maximum of magnitude 64 jn a direction parallel to the z axis.

$${find}\:{the}\:{constant}\:\:{a},{b}\:{and}\:\:{c}\:\:{so} \\ $$$${that}\:{the}\:{direction}\:{derivative}\:{of} \\ $$$$\Phi={axy}^{\mathrm{2}} +{byz}+{cz}^{\mathrm{2}} {x}^{\mathrm{3}} \:{at}\:\left(\mathrm{1},\mathrm{2},−\mathrm{1}\right) \\ $$$${has}\:{a}\:{maximum}\:{of}\:{magnitude} \\ $$$$\mathrm{64}\:{jn}\:{a}\:{direction}\:{parallel}\:{to}\:{the} \\ $$$${z}\:{axis}. \\ $$

Question Number 65788 Answers: 0 Comments: 0

Explicit f(a.b.c)=∫_0 ^(π/2) ((sec(x−a))/(b.cosx + c.sinx)) dx

$${Explicit}\:\:\:{f}\left({a}.{b}.{c}\right)=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{sec}\left({x}−{a}\right)}{{b}.{cosx}\:+\:{c}.{sinx}}\:{dx} \\ $$$$ \\ $$

Question Number 65786 Answers: 0 Comments: 0

Shows that ∣Γ(1+ix)∣^2 =(π/(xsinh(πx))) with Γ(z)=∫_0_ ^∞ t^(z−1) e^(−t) dt Then Prove that ∫_0 ^∞ ∣Γ(1+ix)∣^2 dx =(π/4)

$$\:{Shows}\:{that}\:\:\mid\Gamma\left(\mathrm{1}+{ix}\right)\mid^{\mathrm{2}} =\frac{\pi}{{xsinh}\left(\pi{x}\right)}\:\:\:\:\:\:{with}\:\Gamma\left({z}\right)=\int_{\mathrm{0}_{} } ^{\infty} \:{t}^{{z}−\mathrm{1}} {e}^{−{t}} {dt} \\ $$$${Then}\:{Prove}\:{that}\:\:\int_{\mathrm{0}} ^{\infty} \:\mid\Gamma\left(\mathrm{1}+{ix}\right)\mid^{\mathrm{2}} \:{dx}\:=\frac{\pi}{\mathrm{4}} \\ $$

Question Number 65782 Answers: 0 Comments: 4

Evaluate ∫_0 ^2 (3x^2 −2x + 4)^7 dx hence show that (d/dx)(coshx) = sinh x

$$\:{Evaluate}\:\int_{\mathrm{0}} ^{\mathrm{2}} \left(\mathrm{3}{x}^{\mathrm{2}} −\mathrm{2}{x}\:+\:\mathrm{4}\right)^{\mathrm{7}} {dx} \\ $$$${hence}\:{show}\:{that}\:\:\frac{{d}}{{dx}}\left({coshx}\right)\:=\:{sinh}\:{x} \\ $$

Question Number 65781 Answers: 1 Comments: 0

If xyz ≠ 0 and x+y+z=0 a=10^z b=10^y c=10^x then a^(((1/y)+(1/z))) . b^(((1/z)+(1/x))) .c^(((1/x)+(1/y))) =... a. 0.001 b. 0.01 c. 0.1 d. 1 e. 10

$$\mathrm{If}\:{xyz}\:\neq\:\mathrm{0}\:\mathrm{and}\:{x}+{y}+{z}=\mathrm{0} \\ $$$${a}=\mathrm{10}^{{z}} \\ $$$${b}=\mathrm{10}^{{y}} \\ $$$${c}=\mathrm{10}^{{x}} \\ $$$$\mathrm{then} \\ $$$${a}^{\left(\frac{\mathrm{1}}{{y}}+\frac{\mathrm{1}}{{z}}\right)} .\:{b}^{\left(\frac{\mathrm{1}}{{z}}+\frac{\mathrm{1}}{{x}}\right)} .{c}^{\left(\frac{\mathrm{1}}{{x}}+\frac{\mathrm{1}}{{y}}\right)} =... \\ $$$${a}.\:\mathrm{0}.\mathrm{001} \\ $$$${b}.\:\mathrm{0}.\mathrm{01} \\ $$$${c}.\:\mathrm{0}.\mathrm{1} \\ $$$${d}.\:\mathrm{1} \\ $$$${e}.\:\mathrm{10} \\ $$$$ \\ $$

Question Number 65779 Answers: 0 Comments: 1

calculate lim_(x→0) ((sin(x^2 )−xtan(x))/(1−cos(4x)))

$${calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\:\:\frac{{sin}\left({x}^{\mathrm{2}} \right)−{xtan}\left({x}\right)}{\mathrm{1}−{cos}\left(\mathrm{4}{x}\right)} \\ $$

Question Number 65778 Answers: 0 Comments: 1

find lim_(n→+∞) e^(−n^2 ) (n+1)^(n!)

$${find}\:{lim}_{{n}\rightarrow+\infty} \:{e}^{−{n}^{\mathrm{2}} } \left({n}+\mathrm{1}\right)^{{n}!} \\ $$

Question Number 65777 Answers: 0 Comments: 0

find lim_(n→+∞) e^(−n) ((n+1)!)^n

$${find}\:{lim}_{{n}\rightarrow+\infty} \:\:{e}^{−{n}} \left(\left({n}+\mathrm{1}\right)!\right)^{{n}} \\ $$

Question Number 65776 Answers: 0 Comments: 1

find ∫_(−(π/4)) ^(π/4) ((cosx)/(2+5sinx))dx

$${find}\:\:\int_{−\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{4}}} \:\:\frac{{cosx}}{\mathrm{2}+\mathrm{5}{sinx}}{dx} \\ $$

Question Number 65775 Answers: 0 Comments: 1

calculate ∫_0 ^(2π) ((tanx)/(2+3cosx))dx

$$\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\frac{{tanx}}{\mathrm{2}+\mathrm{3}{cosx}}{dx} \\ $$

Question Number 65774 Answers: 0 Comments: 0

find A_n = ∫_0 ^(2π) ((sin^2 x)/(sin^2 (((nx)/2))))dx (n>0)

$${find}\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\:\:\frac{{sin}^{\mathrm{2}} {x}}{{sin}^{\mathrm{2}} \left(\frac{{nx}}{\mathrm{2}}\right)}{dx}\:\:\:\:\left({n}>\mathrm{0}\right) \\ $$

Question Number 65773 Answers: 0 Comments: 2

let f(x) =∫_0 ^1 (dt/(1+x(√(1+t^2 )))) with x>0 1)detemine a explicit form of f(x) 2)find also g(x) =∫_0 ^1 ((√(1+t^2 ))/((1+x(√(1+t^2 )))^2 ))dt 3) find the value of integrals ∫_0 ^1 (dt/(1+2(√(1+t^2 )))) and ∫_0 ^1 (dt/((1+2(√(1+t^2 )))^2 ))

$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{dt}}{\mathrm{1}+{x}\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }}\:\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right){detemine}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){find}\:{also}\:\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }}{\left(\mathrm{1}+{x}\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }\right)^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{value}\:{of}\:{integrals}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{dt}}{\mathrm{1}+\mathrm{2}\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }}\:{and}\: \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\:\frac{{dt}}{\left(\mathrm{1}+\mathrm{2}\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }\right)^{\mathrm{2}} } \\ $$

Question Number 65771 Answers: 0 Comments: 0

let X_n =∫_0 ^(π/4) sin^n xdx 1) calculate X_0 ,X_1 ,X_2 ,X_3 2) find X_n interms of n 3)find the value of ∫_0 ^(π/4) sin^8 xdx

$${let}\:{X}_{{n}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:{sin}^{{n}} {xdx} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{X}_{\mathrm{0}} \:,{X}_{\mathrm{1}} \:,{X}_{\mathrm{2}} ,{X}_{\mathrm{3}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{X}_{{n}} {interms}\:{of}\:{n} \\ $$$$\left.\mathrm{3}\right){find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:{sin}^{\mathrm{8}} {xdx} \\ $$

Question Number 65770 Answers: 0 Comments: 2

let A_n =∫_0 ^(π/2) cos^n xdx 1) calculate A_0 ,A_2 and A_3 2)calculate A_n interms of n 3) find ∫_0 ^(π/2) cos^8 xdx

$${let}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:{cos}^{{n}} {xdx} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{A}_{\mathrm{0}} ,{A}_{\mathrm{2}} \:{and}\:{A}_{\mathrm{3}} \\ $$$$\left.\mathrm{2}\right){calculate}\:{A}_{{n}} {interms}\:{of}\:{n} \\ $$$$\left.\mathrm{3}\right)\:{find}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:{cos}^{\mathrm{8}} {xdx}\: \\ $$

Question Number 65769 Answers: 0 Comments: 3

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

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

Question Number 65768 Answers: 0 Comments: 1

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

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

Question Number 65767 Answers: 0 Comments: 0

let f(x) =∫_0 ^(+∞) (dt/(t^4 +x^4 )) with x>0 1) determine a explicit form of f(x) 2) find also g(x) =∫_0 ^∞ (dt/((t^4 +x^4 )^2 )) 3)give f^((n)) (x) at form of integral 4) calculate ∫_0 ^∞ (dt/(t^4 +8)) and ∫_0 ^∞ (dt/((t^4 +8)^2 )) 5) calculate A_n =∫_0 ^∞ (dt/((t^4 +x^4 )^n )) with n integr natural

$${let}\:\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{+\infty} \:\:\:\frac{{dt}}{{t}^{\mathrm{4}} +{x}^{\mathrm{4}} }\:\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{also}\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dt}}{\left({t}^{\mathrm{4}} \:+{x}^{\mathrm{4}} \right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{3}\right){give}\:{f}^{\left({n}\right)} \left({x}\right)\:{at}\:{form}\:{of}\:{integral} \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dt}}{{t}^{\mathrm{4}} \:+\mathrm{8}}\:\:{and}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dt}}{\left({t}^{\mathrm{4}} \:+\mathrm{8}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{5}\right)\:{calculate}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dt}}{\left({t}^{\mathrm{4}} \:+{x}^{\mathrm{4}} \right)^{{n}} }\:\:{with}\:{n}\:{integr}\:{natural} \\ $$

Question Number 65763 Answers: 0 Comments: 0