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Question Number 66245 Answers: 1 Comments: 2

prove that ∫e^x dx = e^x + c

$${prove}\:{that} \\ $$$$ \\ $$$$\int{e}^{{x}} \:{dx}\:=\:{e}^{{x}} \:+\:{c} \\ $$

Question Number 66228 Answers: 0 Comments: 3

prove that ∫_2 ^4 ((6x +1)/((2x−3)(3x−2)))dx = ln 10

$${prove}\:{that}\: \\ $$$$\int_{\mathrm{2}} ^{\mathrm{4}} \frac{\mathrm{6}{x}\:+\mathrm{1}}{\left(\mathrm{2}{x}−\mathrm{3}\right)\left(\mathrm{3}{x}−\mathrm{2}\right)}{dx}\:=\:{ln}\:\mathrm{10} \\ $$

Question Number 66227 Answers: 0 Comments: 0

Using a good counter procedure, prove that (∂y/∂x) = lim_(∂x→0) ((f(∂ + x) −f(x))/∂x) for a given function f(x) in x.

$${Using}\:{a}\:{good}\:{counter}\:{procedure},\:{prove}\:{that}\: \\ $$$$\:\:\:\frac{\partial{y}}{\partial{x}}\:=\:\underset{\partial{x}\rightarrow\mathrm{0}} {{lim}}\frac{{f}\left(\partial\:+\:{x}\right)\:−{f}\left({x}\right)}{\partial{x}} \\ $$$${for}\:{a}\:{given}\:{function}\:\:{f}\left({x}\right)\:{in}\:{x}. \\ $$

Question Number 66226 Answers: 0 Comments: 1

the equation f(x)=0 has real roots in the interval (a, b) if A −f(a)>0 and f(b) >0 B f(a) <0 and f(b) <0 C −f(a) >0 and f(b) =0 D f(a) >0 and f(b) < 0

$${the}\:{equation}\:\:{f}\left({x}\right)=\mathrm{0}\:{has}\:{real}\:{roots}\:{in}\: \\ $$$${the}\:{interval}\:\left({a},\:{b}\right)\:{if} \\ $$$${A}\:\:\:\:−{f}\left({a}\right)>\mathrm{0}\:\:{and}\:{f}\left({b}\right)\:>\mathrm{0} \\ $$$${B}\:\:\:{f}\left({a}\right)\:<\mathrm{0}\:{and}\:{f}\left({b}\right)\:<\mathrm{0} \\ $$$${C}\:\:−{f}\left({a}\right)\:>\mathrm{0}\:\:{and}\:{f}\left({b}\right)\:=\mathrm{0} \\ $$$${D}\:\:{f}\left({a}\right)\:>\mathrm{0}\:\:{and}\:{f}\left({b}\right)\:<\:\mathrm{0} \\ $$

Question Number 66225 Answers: 1 Comments: 2

Given that f(x)= { ((−x + 1, x≤ 3_ )),((kx −8, x >3)) :} is continuous then f(5) = A 2 B 0 C −2 D −1

$${Given}\:{that}\:\:\:\:\:{f}\left({x}\right)=\begin{cases}{−{x}\:+\:\mathrm{1},\:\:{x}\leqslant\:\mathrm{3}_{} }\\{{kx}\:−\mathrm{8},\:\:\:\:{x}\:>\mathrm{3}}\end{cases} \\ $$$${is}\:{continuous}\:{then}\:\:{f}\left(\mathrm{5}\right)\:=\: \\ $$$${A}\:\:\:\mathrm{2} \\ $$$${B}\:\:\:\mathrm{0} \\ $$$${C}\:\:−\mathrm{2} \\ $$$${D}\:\:−\mathrm{1} \\ $$$$ \\ $$

Question Number 66216 Answers: 1 Comments: 2

∣a ∣ = 3 ,∣b∣= 5 , a.b =−14 ∣a − b∣ = ?

$$\mid{a}\:\mid\:=\:\mathrm{3}\:,\mid{b}\mid=\:\mathrm{5}\:,\:{a}.{b}\:=−\mathrm{14} \\ $$$$\:\:\mid{a}\:−\:{b}\mid\:=\:? \\ $$

Question Number 66816 Answers: 0 Comments: 3

let x>0 and f(x)=∫_1 ^2 (t+1)(√(t^2 −2xt−1))dt 1) find a explicit form of f(x) 2) determine also g(x)=∫_1 ^2 ((t^2 +t)/(√(t^2 −2xt−1)))dt 3)find the value of integrals ∫_1 ^2 (t+1)(√(t^2 −t−1))dt and ∫_1 ^2 ((t^(2 ) +t)/(√(t^2 −t−1)))dt .

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

Question Number 66815 Answers: 2 Comments: 1

solve the congruence equation 6x ≡ 4 (mod 5) i need help please with some explanations

$${solve}\:{the}\:{congruence}\:{equation}\: \\ $$$$\:\:\mathrm{6}{x}\:\equiv\:\mathrm{4}\:\left({mod}\:\mathrm{5}\right)\:\:{i}\:{need}\:{help}\:{please}\:{with}\:{some}\:{explanations} \\ $$

Question Number 66213 Answers: 0 Comments: 3

calculate A_n =∫_0 ^∞ cos(x^n )dx and B_n =∫_0 ^∞ sin(x^n )dx with n integr and n≥2

$${calculate}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:{cos}\left({x}^{{n}} \right){dx}\:{and}\:{B}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:{sin}\left({x}^{{n}} \right){dx} \\ $$$${with}\:{n}\:{integr}\:{and}\:{n}\geqslant\mathrm{2} \\ $$

Question Number 66211 Answers: 1 Comments: 4