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Question Number 30354 Answers: 1 Comments: 0

If g(x)=∫_0 ^x cos^4 t dt, then g (x+π) =

$$\mathrm{If}\:\:{g}\left({x}\right)=\overset{{x}} {\int}_{\mathrm{0}} \mathrm{cos}^{\mathrm{4}} {t}\:{dt},\:\mathrm{then}\:{g}\:\left({x}+\pi\right)\:= \\ $$

Question Number 30350 Answers: 1 Comments: 0

Question Number 30331 Answers: 2 Comments: 0

Question Number 30340 Answers: 1 Comments: 0

Question Number 30321 Answers: 0 Comments: 3

∫_(−∞) ^∞ (e^(ax) /(e^x +1))dx=?

$$\int_{−\infty} ^{\infty} \frac{\mathrm{e}^{\mathrm{a}{x}} }{\mathrm{e}^{{x}} +\mathrm{1}}{dx}=? \\ $$

Question Number 30323 Answers: 0 Comments: 0

Question Number 30299 Answers: 1 Comments: 5

Question Number 30282 Answers: 0 Comments: 3

Find lim_(n→∞) cos^n (((2π)/n))

$${Find}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}cos}^{{n}} \:\left(\frac{\mathrm{2}\pi}{{n}}\right) \\ $$

Question Number 30348 Answers: 0 Comments: 1

Question Number 30267 Answers: 0 Comments: 7

Can We expand the following expression? (1+x)(1+2x)(1+3x)......(1+nx) or is there any formula for this?

$${Can}\:{We}\:{expand}\:{the}\:{following} \\ $$$${expression}? \\ $$$$\left(\mathrm{1}+{x}\right)\left(\mathrm{1}+\mathrm{2}{x}\right)\left(\mathrm{1}+\mathrm{3}{x}\right)......\left(\mathrm{1}+{nx}\right) \\ $$$${or}\:{is}\:{there}\:{any}\:{formula}\:{for}\:{this}? \\ $$

Question Number 30259 Answers: 1 Comments: 1

Question Number 30258 Answers: 2 Comments: 0

Question Number 30257 Answers: 1 Comments: 5

Question Number 30256 Answers: 0 Comments: 1

Question Number 30245 Answers: 1 Comments: 3

1^2 −3^2 +5^2 −7^2 +•••+17^2 −19^2

$$\mathrm{1}^{\mathrm{2}} −\mathrm{3}^{\mathrm{2}} +\mathrm{5}^{\mathrm{2}} −\mathrm{7}^{\mathrm{2}} +\bullet\bullet\bullet+\mathrm{17}^{\mathrm{2}} −\mathrm{19}^{\mathrm{2}} \\ $$

Question Number 30244 Answers: 0 Comments: 0

if f^(−1) exists and f is differentiable on R ,the f^(−1) is also differentiable.(T/F)

$${if}\:{f}^{−\mathrm{1}} \:{exists}\:{and}\:{f}\:{is}\:{differentiable}\:{on}\:{R}\:,{the}\:{f}^{−\mathrm{1}} \:{is}\:{also}\:{differentiable}.\left({T}/{F}\right) \\ $$$$ \\ $$

Question Number 30235 Answers: 0 Comments: 3

find the sum of the infinite series tan^(−1) ((2/n^2 ))

$${find}\:{the}\:{sum}\:{of}\:{the}\:{infinite} \\ $$$${series}\: \\ $$$$\:\:\:\:\:\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{2}}{{n}^{\mathrm{2}} }\right) \\ $$

Question Number 30233 Answers: 1 Comments: 1

Question Number 30221 Answers: 0 Comments: 0

Question Number 30220 Answers: 0 Comments: 1

let p(x)= x^3 px +q 1) prove that p have double roots⇔ 4p^3 +27q^2 =0 3) let suppose p have 3 real roots differnts prove that 4p^3 +27q^2 <0.

$${let}\:{p}\left({x}\right)=\:{x}^{\mathrm{3}} \:{px}\:+{q} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{p}\:{have}\:{double}\:{roots}\Leftrightarrow\:\mathrm{4}{p}^{\mathrm{3}} \:+\mathrm{27}{q}^{\mathrm{2}} =\mathrm{0} \\ $$$$\left.\mathrm{3}\right)\:{let}\:{suppose}\:{p}\:{have}\:\mathrm{3}\:{real}\:{roots}\:{differnts}\:{prove}\:{that} \\ $$$$\mathrm{4}{p}^{\mathrm{3}} \:+\mathrm{27}{q}^{\mathrm{2}} \:<\mathrm{0}. \\ $$

Question Number 30218 Answers: 0 Comments: 0

prove that D(x^5 −1,x^2 +x+1)=1.

$${prove}\:{that}\:{D}\left({x}^{\mathrm{5}} −\mathrm{1},{x}^{\mathrm{2}} +{x}+\mathrm{1}\right)=\mathrm{1}. \\ $$

Question Number 30217 Answers: 0 Comments: 0

prove that ∀n∈N^★ (1/(√n)) −(1/(√(n+1))) ≥ (1/(2(n+1)(√(n+1)))) 2) prove that u_n = Σ_(k=1) ^n (1/(k(√k))) is convergente .

$${prove}\:{that}\:\:\forall{n}\in{N}^{\bigstar} \:\:\:\:\frac{\mathrm{1}}{\sqrt{{n}}}\:−\frac{\mathrm{1}}{\sqrt{{n}+\mathrm{1}}}\:\geqslant\:\frac{\mathrm{1}}{\mathrm{2}\left({n}+\mathrm{1}\right)\sqrt{{n}+\mathrm{1}}} \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:{u}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\mathrm{1}}{{k}\sqrt{{k}}}\:{is}\:{convergente}\:. \\ $$

Question Number 30216 Answers: 0 Comments: 0

let I(x)= ∫_0 ^π (dt/(x^2 +cos^2 t)) 1) prove that I(x)= 2∫_0 ^(π/2) (dt/(x^2 +cos^2 t)) 2) find the value of I(x).

$${let}\:{I}\left({x}\right)=\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\frac{{dt}}{{x}^{\mathrm{2}} \:+{cos}^{\mathrm{2}} {t}} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{I}\left({x}\right)=\:\mathrm{2}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{dt}}{{x}^{\mathrm{2}} \:+{cos}^{\mathrm{2}} {t}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:{I}\left({x}\right). \\ $$

Question Number 30215 Answers: 0 Comments: 0

let give J(x)= (1/π) ∫_0 ^π cos(xcost)dt 1) find J^′ and J^(′′) in form of integrals 2)prove that J^′ (x)=((−x)/π) ∫_0 ^π sin^2 t cos(xcost)dt and J is solution of d.e. xy^(′′) +y^′ +xy=0

$${let}\:{give}\:{J}\left({x}\right)=\:\frac{\mathrm{1}}{\pi}\:\int_{\mathrm{0}} ^{\pi} {cos}\left({xcost}\right){dt} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{J}^{'} \:{and}\:{J}^{''} \:{in}\:{form}\:{of}\:{integrals} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:{J}^{'} \left({x}\right)=\frac{−{x}}{\pi}\:\int_{\mathrm{0}} ^{\pi} \:{sin}^{\mathrm{2}} {t}\:{cos}\left({xcost}\right){dt}\:{and}\:{J}\:{is} \\ $$$${solution}\:{of}\:{d}.{e}.\:\:{xy}^{''} \:+{y}^{'} \:+{xy}=\mathrm{0} \\ $$

Question Number 30214 Answers: 0 Comments: 1

study the convergence of u_n = Σ_(k=1) ^n (−1)^(k−1) (C_n ^k /k) for that use H_n = Σ_(k=1) ^n (1/k) .

$${study}\:{the}\:{convergence}\:{of}\:{u}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \left(−\mathrm{1}\right)^{{k}−\mathrm{1}} \:\:\frac{{C}_{{n}} ^{{k}} }{{k}} \\ $$$${for}\:{that}\:{use}\:{H}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{k}}\:. \\ $$

Question Number 30213 Answers: 0 Comments: 0

p integr and p≥2 1) prove that ∃c∈ ]0,1[ / ln(ln(p+1))−ln(lnp) =(1/((p+c)ln(p+c))) 2)prove that ln(ln(p+1))−ln(ln(p))<(1/(plnp)) 3) prove that lim_(n→∞) Σ_(k=2) ^n (1/(klnk))=+∞ .

$${p}\:{integr}\:{and}\:{p}\geqslant\mathrm{2} \\ $$$$\left.\mathrm{1}\left.\right)\:{prove}\:{that}\:\exists{c}\in\:\right]\mathrm{0},\mathrm{1}\left[\:/\right. \\ $$$${ln}\left({ln}\left({p}+\mathrm{1}\right)\right)−{ln}\left({lnp}\right)\:=\frac{\mathrm{1}}{\left({p}+{c}\right){ln}\left({p}+{c}\right)} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:{ln}\left({ln}\left({p}+\mathrm{1}\right)\right)−{ln}\left({ln}\left({p}\right)\right)<\frac{\mathrm{1}}{{plnp}} \\ $$$$\left.\mathrm{3}\right)\:{prove}\:{that}\:{lim}_{{n}\rightarrow\infty} \:\sum_{{k}=\mathrm{2}} ^{{n}} \:\frac{\mathrm{1}}{{klnk}}=+\infty\:. \\ $$

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