Question and Answers Forum

1) find f(a) =∫_0 ^∞ (dx/(x^4 +a)) with a>0 2) find g(a)=∫_0 ^∞ (dx/((x^4 +a)^2 )) 3) find value of integrals ∫_0 ^∞ (dx/(x^4 +1)) ,∫_0 ^∞ (dx/(2x^4 +8)) ∫_0 ^∞ (dx/((x^4 +1)^2 )) and ∫_0 ^∞ (dx/((2x^4 +8)^2 ))

$$\left.\mathrm{1}\right)\:{find}\:{f}\left({a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{{x}^{\mathrm{4}} \:+{a}}\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{g}\left({a}\right)=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\left({x}^{\mathrm{4}} \:+{a}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{3}\right)\:{find}\:{value}\:{of}\:{integrals}\:\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{{x}^{\mathrm{4}} \:+\mathrm{1}}\:,\int_{\mathrm{0}} ^{\infty} \:\frac{{dx}}{\mathrm{2}{x}^{\mathrm{4}} \:+\mathrm{8}} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\frac{{dx}}{\left({x}^{\mathrm{4}} \:+\mathrm{1}\right)^{\mathrm{2}} }\:{and}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\left(\mathrm{2}{x}^{\mathrm{4}} +\mathrm{8}\right)^{\mathrm{2}} } \\ $$

Question Number 85158 Answers: 0 Comments: 0

calculate U_n = ∫_(−(1/n)) ^(1/n) x^2 (√((1−x)/(1+x)))dx (n integr and n≥2) 2) find nature of Σ U_n

$${calculate}\:{U}_{{n}} =\:\int_{−\frac{\mathrm{1}}{{n}}} ^{\frac{\mathrm{1}}{{n}}} \:{x}^{\mathrm{2}} \sqrt{\frac{\mathrm{1}−{x}}{\mathrm{1}+{x}}}{dx}\:\:\:\left({n}\:{integr}\:{and}\:{n}\geqslant\mathrm{2}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{nature}\:{of}\:\Sigma\:{U}_{{n}} \\ $$

Question Number 85153 Answers: 1 Comments: 1

∫(√(4x^2 −4))dx = ...

$$\int\sqrt{\mathrm{4}\boldsymbol{\mathrm{x}}^{\mathrm{2}} −\mathrm{4}}\boldsymbol{\mathrm{dx}}\:=\:... \\ $$$$ \\ $$$$ \\ $$

Question Number 85148 Answers: 0 Comments: 0

∫_0 ^1 ((1+x^4 )/(1+x^3 +x^7 )) dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}+{x}^{\mathrm{4}} }{\mathrm{1}+{x}^{\mathrm{3}} +{x}^{\mathrm{7}} }\:{dx} \\ $$

Question Number 85146 Answers: 2 Comments: 1

find minimum & maximum value of function f(x)= −sin^2 x+sin x−(1/2) , −π≤x≤π

$$\mathrm{find}\:\mathrm{minimum}\:\&\:\mathrm{maximum}\:\mathrm{value}\: \\ $$$$\mathrm{of}\:\mathrm{function}\: \\ $$$$\mathrm{f}\left(\mathrm{x}\right)=\:−\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}+\mathrm{sin}\:\mathrm{x}−\frac{\mathrm{1}}{\mathrm{2}}\:,\:−\pi\leqslant\mathrm{x}\leqslant\pi \\ $$

Question Number 85142 Answers: 1 Comments: 0

show that ∫_0 ^n [x^2 ]dx =n(n^2 −1)−Σ_(k=1) ^(n^2 −1) (√k)

$${show}\:{that} \\ $$$$\int_{\mathrm{0}} ^{{n}} \left[{x}^{\mathrm{2}} \right]{dx}\:={n}\left({n}^{\mathrm{2}} −\mathrm{1}\right)−\underset{{k}=\mathrm{1}} {\overset{{n}^{\mathrm{2}} −\mathrm{1}} {\sum}}\sqrt{{k}}\: \\ $$

Question Number 85131 Answers: 0 Comments: 4

what procedure will you use to find the inverse of A = ((2,1,9),(1,5,1),(3,0,3) )

$$\mathrm{what}\:\mathrm{procedure}\:\mathrm{will}\:\mathrm{you}\:\mathrm{use}\:\mathrm{to}\: \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{inverse}\:\mathrm{of} \\ $$$$\:\mathrm{A}\:=\:\begin{pmatrix}{\mathrm{2}}&{\mathrm{1}}&{\mathrm{9}}\\{\mathrm{1}}&{\mathrm{5}}&{\mathrm{1}}\\{\mathrm{3}}&{\mathrm{0}}&{\mathrm{3}}\end{pmatrix} \\ $$

Question Number 85130 Answers: 2 Comments: 0

given f(x)= ((√2)+1)sin x +((√2)−1)cos x find masimum value of function [f(x)]^2

$$\mathrm{given}\:\mathrm{f}\left(\mathrm{x}\right)=\:\left(\sqrt{\mathrm{2}}+\mathrm{1}\right)\mathrm{sin}\:\mathrm{x}\:+\left(\sqrt{\mathrm{2}}−\mathrm{1}\right)\mathrm{cos}\:\mathrm{x} \\ $$$$\mathrm{find}\:\mathrm{masimum}\:\mathrm{value}\:\mathrm{of}\:\mathrm{function} \\ $$$$\left[\mathrm{f}\left(\mathrm{x}\right)\right]^{\mathrm{2}} \\ $$

Question Number 85129 Answers: 0 Comments: 2

lim_(x→e) [∫_0 ^e ((1/x))dx] =?

$$\underset{{x}\rightarrow{e}} {\mathrm{lim}}\:\left[\underset{\mathrm{0}} {\overset{{e}} {\int}}\left(\frac{\mathrm{1}}{{x}}\right){dx}\right]\:=? \\ $$

Question Number 85127 Answers: 1 Comments: 4

evaluate: lim_(x→0) (√x) ln(sin x)

$$\mathrm{evaluate}: \\ $$$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\sqrt{{x}}\:\mathrm{ln}\left(\mathrm{sin}\:{x}\right) \\ $$$$ \\ $$

Question Number 85116 Answers: 0 Comments: 1

Reduce the equations to Clairaut′s form and find the general solution : x^2 p^2 +yp(2x+y)+y^2 =0 (put y=u and xy=v)

$$\:\mathrm{Reduce}\:\mathrm{the}\:\mathrm{equations}\:\mathrm{to}\:\mathrm{Clairaut}'\mathrm{s}\:\mathrm{form} \\ $$$$\:\mathrm{and}\:\mathrm{find}\:\mathrm{the}\:\mathrm{general}\:\mathrm{solution}\:: \\ $$$$\:\boldsymbol{\mathrm{x}}^{\mathrm{2}} \boldsymbol{\mathrm{p}}^{\mathrm{2}} +\boldsymbol{\mathrm{yp}}\left(\mathrm{2}\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}\right)+\boldsymbol{\mathrm{y}}^{\mathrm{2}} =\mathrm{0}\:\:\:\:\:\:\left({put}\:\boldsymbol{{y}}=\boldsymbol{{u}}\:{and}\:\boldsymbol{{xy}}=\boldsymbol{{v}}\right) \\ $$$$\: \\ $$

Question Number 85111 Answers: 1 Comments: 4

Solve the differential equation: ★.(1+x+xy^2 )dy+(y+y^3 )dx

$$\:\boldsymbol{\mathrm{Solve}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{differential}}\:\boldsymbol{\mathrm{equation}}: \\ $$$$\:\bigstar.\left(\mathrm{1}+\mathrm{x}+\mathrm{xy}^{\mathrm{2}} \right)\mathrm{dy}+\left(\mathrm{y}+\mathrm{y}^{\mathrm{3}} \right)\mathrm{dx} \\ $$$$\: \\ $$

Question Number 85105 Answers: 1 Comments: 0

Question Number 85104 Answers: 1 Comments: 2

Given { ((x^2 −2xy−3x = −1)),((4y^2 −2xy+6y = −1)) :} find 2y − x

$$\mathrm{Given}\: \\ $$$$\begin{cases}{\mathrm{x}^{\mathrm{2}} −\mathrm{2xy}−\mathrm{3x}\:=\:−\mathrm{1}}\\{\mathrm{4y}^{\mathrm{2}} −\mathrm{2xy}+\mathrm{6y}\:=\:−\mathrm{1}}\end{cases} \\ $$$$\mathrm{find}\:\mathrm{2y}\:−\:\mathrm{x} \\ $$

Question Number 85103 Answers: 0 Comments: 0

Question Number 85097 Answers: 1 Comments: 0

∫_(−π) ^π x^(2020) (sin x+cos x) dx = 8 find ∫_(−π) ^π x^(2020) cos x dx = ?

$$\underset{−\pi} {\overset{\pi} {\int}}\:\mathrm{x}^{\mathrm{2020}} \:\left(\mathrm{sin}\:\mathrm{x}+\mathrm{cos}\:\mathrm{x}\right)\:\mathrm{dx}\:=\:\mathrm{8} \\ $$$$\mathrm{find}\:\underset{−\pi} {\overset{\pi} {\int}}\:\mathrm{x}^{\mathrm{2020}} \:\mathrm{cos}\:\mathrm{x}\:\mathrm{dx}\:=\:? \\ $$

Question Number 85091 Answers: 0 Comments: 1

Question Number 85088 Answers: 0 Comments: 1

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