Question and Answers Forum

let F(x) = ∫_0 ^(π/2) cos(xsint)dt 1) prove that ∀u ∈R 1−(u^2 /2) ≤cosu≤1−(u^2 /2) +(u^4 /(24)) 2) prove that (π/2)(1−(x^2 /4))≤F(x)≤ (π/2)(1−(x^2 /4) +(x^4 /(64)))

$${let}\:{F}\left({x}\right)\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:{cos}\left({xsint}\right){dt} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\:\forall{u}\:\in{R}\:\:\mathrm{1}−\frac{{u}^{\mathrm{2}} }{\mathrm{2}}\:\leqslant{cosu}\leqslant\mathrm{1}−\frac{{u}^{\mathrm{2}} }{\mathrm{2}}\:+\frac{{u}^{\mathrm{4}} }{\mathrm{24}} \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\frac{\pi}{\mathrm{2}}\left(\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{4}}\right)\leqslant{F}\left({x}\right)\leqslant\:\frac{\pi}{\mathrm{2}}\left(\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{4}}\:+\frac{{x}^{\mathrm{4}} }{\mathrm{64}}\right) \\ $$

Question Number 40150 Answers: 0 Comments: 1

let f_n (x) =(1/((1+x^n )^(1+(1/n)) )) ddfined on [0,1] 1) prove that f_n →^(cs) f (n→+∞) 2) calculate I_n = ∫_0 ^1 f_n (x)dx

$${let}\:{f}_{{n}} \left({x}\right)\:=\frac{\mathrm{1}}{\left(\mathrm{1}+{x}^{{n}} \right)^{\mathrm{1}+\frac{\mathrm{1}}{{n}}} }\:\:\:{ddfined}\:{on}\:\left[\mathrm{0},\mathrm{1}\right] \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{f}_{{n}} \rightarrow^{{cs}} \:{f}\:\left({n}\rightarrow+\infty\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{f}_{{n}} \left({x}\right){dx} \\ $$$$ \\ $$

Question Number 40149 Answers: 0 Comments: 1

let u_n = (1/(√n)) Σ_(k=1) ^n (1/(√(n+4k))) find lim_(n→+∞) u_n

$${let}\:{u}_{{n}} =\:\frac{\mathrm{1}}{\sqrt{{n}}}\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\mathrm{1}}{\sqrt{{n}+\mathrm{4}{k}}} \\ $$$${find}\:{lim}_{{n}\rightarrow+\infty} \:{u}_{{n}} \\ $$

Question Number 40148 Answers: 3 Comments: 0

let f(x)= (x^3 /((1+x^2 )^(3/2) )) 1) calculate ∫_0 ^1 f(x)dx 2) let S_n = (1/n^4 ) Σ_(k=1) ^n (k^3 /(√((1+((k/n))^2 )^3 ))) find lim_(n→+∞) S_n

$${let}\:\:{f}\left({x}\right)=\:\frac{{x}^{\mathrm{3}} }{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\frac{\mathrm{3}}{\mathrm{2}}} } \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{f}\left({x}\right){dx} \\ $$$$\left.\mathrm{2}\right)\:{let}\:\:{S}_{{n}} =\:\frac{\mathrm{1}}{{n}^{\mathrm{4}} }\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\:\:\:\frac{{k}^{\mathrm{3}} }{\sqrt{\left(\mathrm{1}+\left(\frac{{k}}{{n}}\right)^{\mathrm{2}} \right)^{\mathrm{3}} }} \\ $$$${find}\:{lim}_{{n}\rightarrow+\infty} \:\:{S}_{{n}} \\ $$

Question Number 40147 Answers: 0 Comments: 2

calculate ∫_0 ^2 (√(x^3 (2−x)))dx

$${calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{2}} \:\:\sqrt{{x}^{\mathrm{3}} \left(\mathrm{2}−{x}\right)}{dx} \\ $$

Question Number 40146 Answers: 1 Comments: 1

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

$${find}\:\:\:\int_{\frac{\mathrm{1}}{\mathrm{2}}} ^{\mathrm{1}} \:\:\:\:\frac{{dx}}{\sqrt{\mathrm{4}{x}^{\mathrm{2}} \:−\mathrm{1}}\:+\sqrt{\mathrm{4}{x}^{\mathrm{2}} \:+\mathrm{1}}} \\ $$

Question Number 40145 Answers: 1 Comments: 1

calculate ∫_(−7) ^(−3) (((x−1)dx)/(√(x^2 +2x−3)))

$${calculate}\:\int_{−\mathrm{7}} ^{−\mathrm{3}} \:\:\:\frac{\left({x}−\mathrm{1}\right){dx}}{\sqrt{{x}^{\mathrm{2}} \:+\mathrm{2}{x}−\mathrm{3}}} \\ $$

Question Number 40144 Answers: 1 Comments: 0

find ∫_1 ^2 x(√(x^2 −2x +5)) dx

$${find}\:\:\int_{\mathrm{1}} ^{\mathrm{2}} {x}\sqrt{{x}^{\mathrm{2}} \:−\mathrm{2}{x}\:+\mathrm{5}}\:{dx} \\ $$

Question Number 40143 Answers: 0 Comments: 1

find the value of ∫_0 ^(π/4) ((tan(x)dx)/((√2)cos(x) +2sin^2 (x)))

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\:\:\:\frac{{tan}\left({x}\right){dx}}{\sqrt{\mathrm{2}}{cos}\left({x}\right)\:+\mathrm{2}{sin}^{\mathrm{2}} \left({x}\right)} \\ $$

Question Number 40141 Answers: 0 Comments: 1

find ∫_0 ^(π/2) (dx/(3+sinx))

$${find}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\:\:\:\frac{{dx}}{\mathrm{3}+{sinx}} \\ $$

Question Number 40136 Answers: 1 Comments: 0

let A_n = ∫_0 ^1 x^n e^(−x) dx 1) calculate A_1 and A_2 2) prove that A_(n+1) =(n+1)A_n −(1/e) 3) calculate A_3 , A_4 , and A_5 4) calculate I = ∫_0 ^1 (−x^3 +2x^(2 ) −x)e^(−x) dx

$${let}\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{x}^{{n}} \:{e}^{−{x}} {dx} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{A}_{\mathrm{1}} \:\:\:{and}\:{A}_{\mathrm{2}} \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\:\:{A}_{{n}+\mathrm{1}} =\left({n}+\mathrm{1}\right){A}_{{n}} \:−\frac{\mathrm{1}}{{e}} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:{A}_{\mathrm{3}} \:\:\:,\:{A}_{\mathrm{4}} ,\:{and}\:{A}_{\mathrm{5}} \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:{I}\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \left(−{x}^{\mathrm{3}} \:+\mathrm{2}{x}^{\mathrm{2}\:} \:−{x}\right){e}^{−{x}} \:{dx} \\ $$

Question Number 40138 Answers: 0 Comments: 2

let a_k =∫_(−(π/(2 )) +kπ) ^(−(π/2) +(k+1)π) e^(−t) cost dt 1) calculate a_k 2) find lim_(n→+∞) Σ_(k=0) ^n ∣a_k ∣.

$${let}\:\:{a}_{{k}} \:\:\:=\int_{−\frac{\pi}{\mathrm{2}\:}\:+{k}\pi} ^{−\frac{\pi}{\mathrm{2}}\:+\left({k}+\mathrm{1}\right)\pi} \:{e}^{−{t}} \:{cost}\:{dt} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{a}_{{k}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow+\infty} \:\:\sum_{{k}=\mathrm{0}} ^{{n}} \:\:\mid{a}_{{k}} \mid. \\ $$

Question Number 40134 Answers: 0 Comments: 1

calculate ∫_1 ^2 (x^3 /((1+x^4 )^2 ))dx

$${calculate}\:\:\int_{\mathrm{1}} ^{\mathrm{2}} \:\:\:\frac{{x}^{\mathrm{3}} }{\left(\mathrm{1}+{x}^{\mathrm{4}} \right)^{\mathrm{2}} }{dx} \\ $$

Question Number 40133 Answers: 0 Comments: 1

find ∫_0 ^1 (dt/((1+t^2 )^2 ))

$${find}\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{dt}}{\left(\mathrm{1}+{t}^{\mathrm{2}} \right)^{\mathrm{2}} } \\ $$

Question Number 40132 Answers: 0 Comments: 1

calculate ∫_0 ^1 (dt/((1+t^2 )^3 ))