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Question Number 225629 Answers: 2 Comments: 0

Question Number 225391 Answers: 0 Comments: 0

Σ_(n = 1) ^∞ (− 1)^(n − 1) ((H_n H_(2n) ^((2)) )/n) = ?

$$\:\:\:\:\:\:\:\underset{\mathrm{n}\:\:\:=\:\:\:\mathrm{1}} {\overset{\infty} {\sum}}\:\left(−\:\:\:\:\mathrm{1}\right)^{\mathrm{n}\:\:\:\:−\:\:\:\mathrm{1}} \:\frac{\mathrm{H}_{\mathrm{n}} \:\mathrm{H}_{\mathrm{2n}} ^{\left(\mathrm{2}\right)} }{\mathrm{n}}\:\:\:\:\:\:\:=\:\:\:\:\:? \\ $$

Question Number 225382 Answers: 1 Comments: 0

Been a while guys ∫_0 ^( 1) ((xln(1+x))/(1+x^2 ))dx

$$\mathrm{Been}\:\mathrm{a}\:\mathrm{while}\:\mathrm{guys} \\ $$$$\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\mathrm{xln}\left(\mathrm{1}+\mathrm{x}\right)}{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\mathrm{dx} \\ $$

Question Number 224920 Answers: 0 Comments: 0

∫_( 0) ^( 1) ((x ln(1 + x) Li_2 (x))/(1 + x^2 )) dx

$$\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\frac{\mathrm{x}\:\mathrm{ln}\left(\mathrm{1}\:\:\:+\:\:\:\mathrm{x}\right)\:\mathrm{Li}_{\mathrm{2}} \left(\mathrm{x}\right)}{\mathrm{1}\:\:\:\:+\:\:\:\:\mathrm{x}^{\mathrm{2}} }\:\mathrm{dx} \\ $$

Question Number 224839 Answers: 0 Comments: 0

∫_( 0) ^( 1) ((x tan^(− 1) (x) ln(1 − x))/(1 + x^2 )) dx

$$\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\frac{\mathrm{x}\:\mathrm{tan}^{−\:\mathrm{1}} \left(\mathrm{x}\right)\:\mathrm{ln}\left(\mathrm{1}\:\:\:−\:\:\:\mathrm{x}\right)}{\mathrm{1}\:\:\:+\:\:\:\mathrm{x}^{\mathrm{2}} }\:\mathrm{dx} \\ $$

Question Number 224798 Answers: 0 Comments: 0

∫_( 0) ^( 1) ((arctan^2 (x) ln(1 − x))/x^2 ) dx

$$\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\frac{\mathrm{arctan}^{\mathrm{2}} \left(\mathrm{x}\right)\:\mathrm{ln}\left(\mathrm{1}\:\:\:−\:\:\:\mathrm{x}\right)}{\mathrm{x}^{\mathrm{2}} }\:\mathrm{dx} \\ $$

Question Number 224771 Answers: 0 Comments: 0

K=∫_0 ^( ∞) ((sinx)/(coshx)) (e^(−2x) −e^(−4x) )dx=?

$$ \\ $$$$\:\:{K}=\int_{\mathrm{0}} ^{\:\infty} \frac{{sinx}}{{coshx}}\:\left({e}^{−\mathrm{2}{x}} −{e}^{−\mathrm{4}{x}} \right){dx}=?\:\:\:\:\: \\ $$$$\:\: \\ $$

Question Number 224505 Answers: 0 Comments: 0

∫_0 ^1 e^(ax) J_0 (ȷ_(0m) x)dx Where J_0 is the Bessel function and ȷ_(0m) its m-th zero

$$\int_{\mathrm{0}} ^{\mathrm{1}} {e}^{{ax}} {J}_{\mathrm{0}} \left(\jmath_{\mathrm{0}{m}} {x}\right){dx} \\ $$$$\mathrm{Where}\:{J}_{\mathrm{0}} \:\mathrm{is}\:\mathrm{the}\:\mathrm{Bessel}\:\mathrm{function}\:\mathrm{and}\:\jmath_{\mathrm{0}{m}} \:\mathrm{its}\:{m}-\mathrm{th}\:\mathrm{zero} \\ $$

Question Number 224373 Answers: 0 Comments: 0

Show that ; I = ∫_( 0) ^( 1) ∫_( 0) ^( 1) ((ln(1+(√(xy))) ln(1+ (√((1−x)/(1−y)))))/( (√(1−x)) (√(1−y)) (x+y))) dxdy I = ζ(3)−((70)/(351))−((280)/(351)) ln 2−((40)/(117)) ln^2 2 +((412)/(351)) ln^3 2 + ((167)/(2106)) π^2 ln 2

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{Show}}\:\boldsymbol{\mathrm{that}}\:; \\ $$$$\:\:\:\:\:\mathcal{I}\:=\:\underset{\:\:\mathrm{0}} {\overset{\:\:\mathrm{1}} {\int}}\underset{\:\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:\:\frac{\mathrm{ln}\left(\mathrm{1}+\sqrt{{xy}}\right)\:\mathrm{ln}\left(\mathrm{1}+\:\sqrt{\frac{\mathrm{1}−{x}}{\mathrm{1}−{y}}}\right)}{\:\sqrt{\mathrm{1}−{x}}\:\:\sqrt{\mathrm{1}−{y}}\:\:\left({x}+{y}\right)}\:\:{dxdy}\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\mathcal{I}\:=\:\zeta\left(\mathrm{3}\right)−\frac{\mathrm{70}}{\mathrm{351}}−\frac{\mathrm{280}}{\mathrm{351}}\:\mathrm{ln}\:\mathrm{2}−\frac{\mathrm{40}}{\mathrm{117}}\:\mathrm{ln}^{\mathrm{2}} \:\mathrm{2}\:+\frac{\mathrm{412}}{\mathrm{351}}\:\mathrm{ln}^{\mathrm{3}} \:\mathrm{2}\:+\:\frac{\mathrm{167}}{\mathrm{2106}}\:\pi^{\mathrm{2}} \:\mathrm{ln}\:\mathrm{2}\:\:\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$

Question Number 224248 Answers: 0 Comments: 0

∫_0 ^1 ((ln(1−x)Li_2 (1−(√x)))/x) dx

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{\mathrm{ln}\left(\mathrm{1}−\mathrm{x}\right)\mathrm{Li}_{\mathrm{2}} \left(\mathrm{1}−\sqrt{\mathrm{x}}\right)}{\mathrm{x}}\:\:\mathrm{dx} \\ $$$$ \\ $$

Question Number 224201 Answers: 0 Comments: 0

∫ (x^3 /(x^7 −8x^2 )) dx

$$\int\:\frac{\mathrm{x}^{\mathrm{3}} }{\mathrm{x}^{\mathrm{7}} −\mathrm{8x}^{\mathrm{2}} }\:\mathrm{dx} \\ $$

Question Number 224182 Answers: 1 Comments: 0

Calculate I=∫ ((sin x)/(1+ sin x)) dx

$$\mathrm{Calculate} \\ $$$${I}=\int\:\frac{\mathrm{sin}\:{x}}{\mathrm{1}+\:\mathrm{sin}\:{x}}\:\mathrm{d}{x} \\ $$

Question Number 224085 Answers: 2 Comments: 0

If f(x)=4x^3 +3x^2 +x, Then solve for a and b: max_(x∈R) {∫_x ^2 f(t)dt}=a where x=b

$$\mathrm{If}\:{f}\left({x}\right)=\mathrm{4}{x}^{\mathrm{3}} +\mathrm{3}{x}^{\mathrm{2}} +{x},\:\mathrm{Then}\:\mathrm{solve}\:\mathrm{for}\:{a}\:\mathrm{and}\:{b}: \\ $$$$\underset{{x}\in\mathbb{R}} {\mathrm{max}}\left\{\int_{{x}} ^{\mathrm{2}} {f}\left({t}\right){dt}\right\}={a}\:\mathrm{where}\:{x}={b} \\ $$

Question Number 223958 Answers: 0 Comments: 0

∫_0 ^∞ ((x sinh(x))/(1+cosh^2 (x))) dx

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\infty} \:\frac{{x}\:\mathrm{sinh}\left({x}\right)}{\mathrm{1}+\mathrm{cosh}^{\mathrm{2}} \left({x}\right)}\:\mathrm{d}{x} \\ $$$$ \\ $$

Question Number 223933 Answers: 1 Comments: 0

I=∫_0 ^1 ((ln(x+1))/(x^2 +1))dx

$${I}=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{ln}\left({x}+\mathrm{1}\right)}{{x}^{\mathrm{2}} +\mathrm{1}}{dx} \\ $$

Question Number 223923 Answers: 1 Comments: 0

find ∫_0 ^∞ ((ln(1+x))/(1+x^3 ))dx

$${find}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{ln}\left(\mathrm{1}+{x}\right)}{\mathrm{1}+{x}^{\mathrm{3}} }{dx} \\ $$

Question Number 223920 Answers: 1 Comments: 0

find ∫_0 ^(π/2) (x^2 /(sin^2 x))dx

$${find}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{{x}^{\mathrm{2}} }{{sin}^{\mathrm{2}} {x}}{dx} \\ $$

Question Number 223786 Answers: 0 Comments: 4

Evaluate ; ∫ (√( tan x)) dx , Using feynman′s trick

$$ \\ $$$$\:\:\:\:\boldsymbol{\mathrm{Evaluate}}\:;\:\int\:\sqrt{\:\boldsymbol{\mathrm{tan}}\:\boldsymbol{{x}}}\:\boldsymbol{\mathrm{d}{x}}\:,\:\boldsymbol{\mathrm{Using}}\:\boldsymbol{\mathrm{feynman}}'\boldsymbol{\mathrm{s}}\:\boldsymbol{\mathrm{trick}} \\ $$$$ \\ $$

Question Number 223728 Answers: 1 Comments: 0

∫_0_ ^1 ((ln(1+(√x))∙ln(1+x))/(1+(√x))) dx

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}_{\:} } ^{\mathrm{1}} \:\frac{\mathrm{ln}\left(\mathrm{1}+\sqrt{{x}}\right)\centerdot\mathrm{ln}\left(\mathrm{1}+{x}\right)}{\mathrm{1}+\sqrt{{x}}}\:\mathrm{d}{x} \\ $$

Question Number 223631 Answers: 2 Comments: 1

Question Number 223580 Answers: 3 Comments: 0

∫_0 ^(1 ) ((e^(−r^2 ) sin(1/r^2 )ln(r+1))/r^2 ) dr

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}\:} \:\frac{{e}^{−\boldsymbol{{r}}^{\mathrm{2}} } \boldsymbol{\mathrm{sin}}\left(\mathrm{1}/\boldsymbol{{r}}^{\mathrm{2}} \right)\boldsymbol{\mathrm{ln}}\left(\boldsymbol{{r}}+\mathrm{1}\right)}{\boldsymbol{{r}}^{\mathrm{2}} }\:\boldsymbol{\mathrm{d}{r}} \\ $$$$ \\ $$

Question Number 223534 Answers: 1 Comments: 0

∫_( 0) ^( 1) ((ln(x))/x) ln^3 (((1 − x)/(1 + x))) dx

$$\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\frac{\mathrm{ln}\left(\mathrm{x}\right)}{\mathrm{x}}\:\mathrm{ln}^{\mathrm{3}} \left(\frac{\mathrm{1}\:\:−\:\:\mathrm{x}}{\mathrm{1}\:\:+\:\:\mathrm{x}}\right)\:\mathrm{dx} \\ $$

Question Number 223525 Answers: 0 Comments: 0

I = ∫_0 ^(2π) ∫_0 ^(2π) ∫_0 ^(2π) ∣ cos x + cos y + cos z ∣ dxdydz

$$ \\ $$$$\:\:\:\:{I}\:=\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \int_{\mathrm{0}} ^{\mathrm{2}\pi} \int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\mid\:\mathrm{cos}\:{x}\:+\:\mathrm{cos}\:{y}\:+\:\mathrm{cos}\:{z}\:\:\mid\:\:{dxdydz}\:\:\:\:\:\:\:\: \\ $$$$ \\ $$

Question Number 223461 Answers: 1 Comments: 0

Question Number 223368 Answers: 1 Comments: 0

∫_0 ^1 ln(((2 cos(x^2 ) + ln^2 (x/2))/(1 + cos (x/2)))) dx

$$ \\ $$$$\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\boldsymbol{\mathrm{ln}}\left(\frac{\mathrm{2}\:\boldsymbol{\mathrm{cos}}\left({x}^{\mathrm{2}} \right)\:+\:\boldsymbol{\mathrm{ln}}^{\mathrm{2}} \left({x}/\mathrm{2}\right)}{\mathrm{1}\:+\:\boldsymbol{\mathrm{cos}}\:\left({x}/\mathrm{2}\right)}\right)\:\boldsymbol{\mathrm{d}}{x} \\ $$$$ \\ $$

Question Number 223367 Answers: 0 Comments: 1

∫_0 ^1 ln(2 cos(x^2 ) + ln^2 ((x/2)) dx

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\mathrm{ln}\left(\mathrm{2}\:\mathrm{cos}\left({x}^{\mathrm{2}} \right)\:+\:\mathrm{ln}^{\mathrm{2}} \:\left(\frac{{x}}{\mathrm{2}}\right)\:\mathrm{d}{x}\right. \\ $$$$ \\ $$

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