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Question Number 73545 Answers: 1 Comments: 2

evaluate ∫lnx dx

$${evaluate}\:\:\int{lnx}\:{dx} \\ $$

Question Number 73544 Answers: 1 Comments: 0

expressf(θ)= 8cosθ −15sinθ in the form rcos(θ + α), where r>0 and α is a positive acute angle hence find the general solution of the equation 80cos θ −150sinθ = 13 the maximum and minimum value of (5/(f(θ) + 3))

$${expressf}\left(\theta\right)=\:\mathrm{8}{cos}\theta\:−\mathrm{15}{sin}\theta\:{in}\:{the}\:{form} \\ $$$$\:{rcos}\left(\theta\:+\:\alpha\right),\:{where}\:{r}>\mathrm{0}\:{and}\:\alpha\:{is}\:{a}\:{positive}\:{acute}\:{angle} \\ $$$${hence} \\ $$$${find}\:{the}\:{general}\:{solution}\:{of}\:{the}\:{equation} \\ $$$$\:\:\mathrm{80}{cos}\:\theta\:−\mathrm{150}{sin}\theta\:=\:\mathrm{13} \\ $$$${the}\:{maximum}\:{and}\:{minimum}\:{value}\:{of}\:\:\frac{\mathrm{5}}{{f}\left(\theta\right)\:+\:\mathrm{3}} \\ $$

Question Number 73542 Answers: 0 Comments: 0

Given that the function f(x)= x^3 is differentiable in the interval (−2,2), Use the mean value theorem to find the value of x for which the tangent to the curve is parallel to the chord through the point (−2,8) and (2,8)

$${Given}\:{that}\:{the}\:{function}\:{f}\left({x}\right)=\:{x}^{\mathrm{3}} \:{is}\:{differentiable} \\ $$$${in}\:{the}\:{interval}\:\left(−\mathrm{2},\mathrm{2}\right),\:{Use}\:{the}\:{mean}\:{value}\:{theorem} \\ $$$${to}\:{find}\:{the}\:{value}\:{of}\:{x}\:{for}\:{which}\:{the}\:{tangent}\:{to}\:{the}\:{curve} \\ $$$${is}\:{parallel}\:{to}\:{the}\:{chord}\:{through}\:{the}\:{point}\:\left(−\mathrm{2},\mathrm{8}\right)\:{and}\:\left(\mathrm{2},\mathrm{8}\right) \\ $$

Question Number 73538 Answers: 2 Comments: 0

find the general solution of sin4x + cos2x = 0

$${find}\:{the}\:{general}\:{solution}\:{of}\: \\ $$$$\:{sin}\mathrm{4}{x}\:+\:{cos}\mathrm{2}{x}\:=\:\mathrm{0} \\ $$

Question Number 73537 Answers: 1 Comments: 0

prove by induction that 4^n + 3^n +2 is a multiple of 3 ∀ n Z^+

$${prove}\:{by}\:{induction}\:{that}\:\mathrm{4}^{{n}} \:+\:\mathrm{3}^{{n}} \:+\mathrm{2}\:{is}\:{a}\:{multiple}\:{of}\:\mathrm{3} \\ $$$$\forall\:{n}\:{Z}^{+} \\ $$

Question Number 73536 Answers: 1 Comments: 0

prove that they are infinitely many primes

$${prove}\:{that}\:{they}\:{are}\:{infinitely}\:{many}\:{primes} \\ $$

Question Number 73530 Answers: 1 Comments: 0

given the 3^(rd) degree polynomial P(x) = (2x −1)(x−3)Q(x) + 12x−8 given that (x−1) is a factor of P(x) and P(0) = 10 find Q(x)

$${given}\:{the}\:\mathrm{3}^{{rd}} \:{degree}\:\:{polynomial} \\ $$$${P}\left({x}\right)\:=\:\left(\mathrm{2}{x}\:−\mathrm{1}\right)\left({x}−\mathrm{3}\right){Q}\left({x}\right)\:+\:\mathrm{12}{x}−\mathrm{8} \\ $$$${given}\:{that}\:\left({x}−\mathrm{1}\right)\:{is}\:{a}\:{factor}\:{of}\:{P}\left({x}\right)\:{and}\:\:{P}\left(\mathrm{0}\right)\:=\:\mathrm{10} \\ $$$${find}\:{Q}\left({x}\right) \\ $$

Question Number 73525 Answers: 1 Comments: 2

Question Number 73523 Answers: 1 Comments: 0

prove that : 2^π >8

$$\mathrm{prove}\:\mathrm{that}\::\: \\ $$$$\:\mathrm{2}^{\pi} >\mathrm{8} \\ $$

Question Number 73518 Answers: 1 Comments: 1

Question Number 73503 Answers: 1 Comments: 2

Question Number 73619 Answers: 0 Comments: 16

Question Number 73499 Answers: 0 Comments: 2

Question Number 73495 Answers: 1 Comments: 0

(d^2 y/dx^2 )=c^2 x^2 y (y=a , x=0 )

$$\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }={c}^{\mathrm{2}} {x}^{\mathrm{2}} {y}\:\:\:\:\:\left({y}={a}\:,\:{x}=\mathrm{0}\:\right) \\ $$

Question Number 73491 Answers: 0 Comments: 1

calculate ∫_(−∞) ^(+∞) e^(−3x^2 −2x) dx

$${calculate}\:\int_{−\infty} ^{+\infty} \:\:{e}^{−\mathrm{3}{x}^{\mathrm{2}} −\mathrm{2}{x}} \:{dx} \\ $$

Question Number 73490 Answers: 0 Comments: 0

calculate f(a) =∫_0 ^∞ e^(−(x^2 +(a/x^2 ))) dx with a>0

$${calculate}\:{f}\left({a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−\left({x}^{\mathrm{2}} \:+\frac{{a}}{{x}^{\mathrm{2}} }\right)} {dx}\:\:{with}\:{a}>\mathrm{0} \\ $$

Question Number 73489 Answers: 0 Comments: 0

calculate ∫_0 ^∞ ((arctan(3+x^2 ))/((2 x^2 +9)^2 ))dx

$${calculate}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{arctan}\left(\mathrm{3}+{x}^{\mathrm{2}} \right)}{\left(\mathrm{2}\:{x}^{\mathrm{2}} +\mathrm{9}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 73488 Answers: 2 Comments: 1

solve xy^(′′) +(x^2 −1)y^′ =x e^(−x^2 )

$${solve}\:\:\:{xy}^{''} \:\:+\left({x}^{\mathrm{2}} −\mathrm{1}\right){y}^{'} \:\:={x}\:{e}^{−{x}^{\mathrm{2}} } \\ $$

Question Number 73487 Answers: 0 Comments: 1

let α and β roots of the equation x^2 −x+2=0 simplify A_p = α^p +β^p and calculate Σ_(p=0) ^(n−1) A_p and Σ_(p=0) ^(n−1) A_p ^2

$${let}\:\:\:\:\alpha\:{and}\:\beta\:{roots}\:{of}\:\:{the}\:{equation}\:\:{x}^{\mathrm{2}} −{x}+\mathrm{2}=\mathrm{0} \\ $$$${simplify}\:\:\:{A}_{{p}} =\:\alpha^{{p}} \:+\beta^{{p}} \:{and}\:{calculate} \\ $$$$\sum_{{p}=\mathrm{0}} ^{{n}−\mathrm{1}} \:{A}_{{p}} \:\:{and}\:\sum_{{p}=\mathrm{0}} ^{{n}−\mathrm{1}} \:\:{A}_{{p}} ^{\mathrm{2}} \\ $$

Question Number 73486 Answers: 0 Comments: 2

let P(x)=(1+ix)^n −(1−ix)^n with n integr decompose the Fraction F (x)=(1/(P(x)))

$${let}\:{P}\left({x}\right)=\left(\mathrm{1}+{ix}\right)^{{n}} −\left(\mathrm{1}−{ix}\right)^{{n}} \:{with}\:{n}\:{integr} \\ $$$${decompose}\:{the}\:{Fraction}\:{F}\:\left({x}\right)=\frac{\mathrm{1}}{{P}\left({x}\right)} \\ $$

Question Number 73485 Answers: 0 Comments: 0

find the roots of P(x)=(1+ix +jx^2 )^n −1 with j =e^(i((2π)/3)) then factorize P(x) inside C[x] decompose the fraction F=(1/P)

$${find}\:{the}\:{roots}\:{of}\:{P}\left({x}\right)=\left(\mathrm{1}+{ix}\:+{jx}^{\mathrm{2}} \right)^{{n}} −\mathrm{1} \\ $$$${with}\:{j}\:={e}^{{i}\frac{\mathrm{2}\pi}{\mathrm{3}}} \:\:\:{then}\:{factorize}\:{P}\left({x}\right)\:{inside}\:{C}\left[{x}\right] \\ $$$${decompose}\:{the}\:{fraction}\:{F}=\frac{\mathrm{1}}{{P}} \\ $$

Question Number 73484 Answers: 1 Comments: 0

decompose inside C(x) the fraction F(x)=(1/((x^2 +1)^n )) calculate ∫_0 ^∞ F(x)dx

$${decompose}\:{inside}\:{C}\left({x}\right)\:{the}\:{fraction} \\ $$$${F}\left({x}\right)=\frac{\mathrm{1}}{\left({x}^{\mathrm{2}} \:+\mathrm{1}\right)^{{n}} } \\ $$$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:{F}\left({x}\right){dx} \\ $$

Question Number 73483 Answers: 1 Comments: 0

find ∫ (dx/(x+2−(√(x^2 −x +7))))

$${find}\:\int\:\:\:\:\frac{{dx}}{{x}+\mathrm{2}−\sqrt{{x}^{\mathrm{2}} −{x}\:+\mathrm{7}}} \\ $$

Question Number 73482 Answers: 1 Comments: 1

find ∫ (dx/((√(x^2 +1))+(√(x^2 +3))))

$${find}\:\int\:\:\:\:\frac{{dx}}{\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}+\sqrt{{x}^{\mathrm{2}} \:+\mathrm{3}}} \\ $$

Question Number 73481 Answers: 0 Comments: 0

find ∫ ln(x−cosx)dx

$${find}\:\int\:\:{ln}\left({x}−{cosx}\right){dx} \\ $$

Question Number 73480 Answers: 0 Comments: 0

find ∫_0 ^∞ xe^(−x^2 ) arcran(x+(1/x))dx

$${find}\:\int_{\mathrm{0}} ^{\infty} \:\:{xe}^{−{x}^{\mathrm{2}} } \:{arcran}\left({x}+\frac{\mathrm{1}}{{x}}\right){dx} \\ $$

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