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IntegrationQuestion and Answers: Page 134

Question Number 118491 Answers: 1 Comments: 0

... ⧫Advanced Calculus⧫... Evaluate:: Ω = ∫_0 ^( ∞) ((secθ)/( (√(4tan^2 θ+5))))dθ ...♠L𝛗rD ∅sE♠... ...♣GooD LucK♣

$$ \\ $$$$...\:\blacklozenge\mathrm{Advanced}\:\mathrm{Calculus}\blacklozenge... \\ $$$$ \\ $$$$\mathrm{Evaluate}:: \\ $$$$ \\ $$$$\Omega\:=\:\int_{\mathrm{0}} ^{\:\infty} \frac{\mathrm{sec}\theta}{\:\sqrt{\mathrm{4tan}^{\mathrm{2}} \theta+\mathrm{5}}}\mathrm{d}\theta \\ $$$$ \\ $$$$...\spadesuit\boldsymbol{\mathrm{L}\phi\mathrm{rD}}\:\boldsymbol{\varnothing\mathrm{sE}}\spadesuit... \\ $$$$ \\ $$$$...\clubsuit\boldsymbol{\mathrm{GooD}}\:\boldsymbol{\mathrm{LucK}}\clubsuit \\ $$

Question Number 118478 Answers: 0 Comments: 0

find ∫_0 ^∞ e^(−2x) ln(1+3x)dx

$$\mathrm{find}\:\int_{\mathrm{0}} ^{\infty} \:\mathrm{e}^{−\mathrm{2x}} \mathrm{ln}\left(\mathrm{1}+\mathrm{3x}\right)\mathrm{dx} \\ $$

Question Number 118476 Answers: 0 Comments: 0

find ∫_(−∞) ^∞ ((arctan(1+2x))/(x^2 +1))dx

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

Question Number 118475 Answers: 0 Comments: 0

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

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

Question Number 118438 Answers: 0 Comments: 0

... nice calculus... evaluate :: lim_(s→0) ((ζ( 1+s )+ζ(1−s))/2) =^? γ γ: euler−mascheroni constant m.n.1970.

$$\:\:\:\:\:\:...\:\:{nice}\:\:{calculus}... \\ $$$$ \\ $$$$\:\:\:\:\:{evaluate}\::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$${lim}_{{s}\rightarrow\mathrm{0}} \frac{\zeta\left(\:\mathrm{1}+{s}\:\right)+\zeta\left(\mathrm{1}−{s}\right)}{\mathrm{2}}\:\overset{?} {=}\gamma \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\gamma:\:{euler}−{mascheroni}\:{constant} \\ $$$$\:\:\:{m}.{n}.\mathrm{1970}. \\ $$$$ \\ $$

Question Number 118436 Answers: 4 Comments: 0

∫ cos^4 (x) cos^4 (2x) dx

$$\:\:\int\:\mathrm{cos}\:^{\mathrm{4}} \left({x}\right)\:\mathrm{cos}\:^{\mathrm{4}} \left(\mathrm{2}{x}\right)\:{dx}\: \\ $$

Question Number 118419 Answers: 4 Comments: 1

∫ ((2sin 2x)/(4cos x+sin 2x)) dx

$$\:\:\:\int\:\frac{\mathrm{2sin}\:\mathrm{2}{x}}{\mathrm{4cos}\:{x}+\mathrm{sin}\:\mathrm{2}{x}}\:{dx}\: \\ $$

Question Number 118395 Answers: 2 Comments: 5

∫_a ^b ((f(x))/(f(x)+f(a+b−x)))dx

$$\int_{{a}} ^{{b}} \frac{{f}\left({x}\right)}{{f}\left({x}\right)+{f}\left({a}+{b}−{x}\right)}{dx} \\ $$

Question Number 207620 Answers: 1 Comments: 0

prove that ∫_(−a) ^a (dx/(x^n +1+(√(x^(2n) +1))))=a

$$\mathrm{prove}\:\mathrm{that}\:\underset{−{a}} {\overset{{a}} {\int}}\:\frac{{dx}}{{x}^{{n}} +\mathrm{1}+\sqrt{{x}^{\mathrm{2}{n}} +\mathrm{1}}}={a} \\ $$

Question Number 118347 Answers: 1 Comments: 0

∫ (dx/( (√x) +(x)^(1/(3 )) ))

$$\:\:\:\int\:\frac{{dx}}{\:\sqrt{{x}}\:+\sqrt[{\mathrm{3}\:}]{{x}}}\: \\ $$

Question Number 118340 Answers: 1 Comments: 1

∫_(π/3) ^(π/2) (dx/(1+sin x−cos x))

$$\:\:\:\underset{\pi/\mathrm{3}} {\overset{\pi/\mathrm{2}} {\int}}\:\frac{{dx}}{\mathrm{1}+\mathrm{sin}\:{x}−\mathrm{cos}\:{x}} \\ $$

Question Number 118338 Answers: 3 Comments: 0

solve ∫ (dx/(3−5sin x))

$$\:\:\:\:{solve}\:\int\:\frac{{dx}}{\mathrm{3}−\mathrm{5sin}\:{x}}\: \\ $$

Question Number 118318 Answers: 1 Comments: 0

Question Number 118307 Answers: 2 Comments: 1

∫^∞ _0 ((x^2 −2)/(x^4 +x^2 +1)) dx

$$\:\:\:\underset{\mathrm{0}} {\int}^{\infty} \:\frac{{x}^{\mathrm{2}} −\mathrm{2}}{{x}^{\mathrm{4}} +{x}^{\mathrm{2}} +\mathrm{1}}\:{dx} \\ $$

Question Number 118292 Answers: 2 Comments: 0

Prove that: ∫_0 ^( 1) ((x^n −1)/(lnx)) = ln∣n+1∣

$$\boldsymbol{\mathrm{P}}\mathrm{rove}\:\mathrm{that}: \\ $$$$\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\mathrm{x}^{\mathrm{n}} −\mathrm{1}}{\mathrm{lnx}}\:=\:\boldsymbol{\mathrm{ln}}\mid\boldsymbol{\mathrm{n}}+\mathrm{1}\mid \\ $$

Question Number 118270 Answers: 0 Comments: 1

∫x a^x (1−a)^(1−x) dx?

$$\:\:\:\int{x}\:{a}^{{x}} \:\left(\mathrm{1}−{a}\right)^{\mathrm{1}−{x}} \:{dx}? \\ $$

Question Number 118278 Answers: 1 Comments: 0

Evaluate ∫_( 0) ^( (π/3)) tan^2 xsec((x/3))dx ★

$$\mathrm{Evaluate} \\ $$$$\int_{\:\mathrm{0}} ^{\:\frac{\pi}{\mathrm{3}}} \mathrm{tan}^{\mathrm{2}} \mathrm{xsec}\left(\frac{\mathrm{x}}{\mathrm{3}}\right)\mathrm{dx} \\ $$$$\bigstar \\ $$

Question Number 118246 Answers: 1 Comments: 0

∫_0 ^1 ((ln x)/(x+1)) dx =?

$$\:\:\:\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\frac{\mathrm{ln}\:{x}}{{x}+\mathrm{1}}\:{dx}\:=? \\ $$

Question Number 118230 Answers: 1 Comments: 0

If ∫ ((((√x))^5 )/(((√x))^7 +x^6 )) dx = p ln ((x^q /(x^q +1))) + C find the value of p and q.

$$\mathrm{If}\:\int\:\frac{\left(\sqrt{\mathrm{x}}\right)^{\mathrm{5}} }{\left(\sqrt{\mathrm{x}}\right)^{\mathrm{7}} +\mathrm{x}^{\mathrm{6}} }\:\mathrm{dx}\:=\:\mathrm{p}\:\mathrm{ln}\:\left(\frac{\mathrm{x}^{\mathrm{q}} }{\mathrm{x}^{\mathrm{q}} +\mathrm{1}}\right)\:+\:\mathrm{C}\: \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{p}\:\mathrm{and}\:\mathrm{q}. \\ $$

Question Number 118218 Answers: 3 Comments: 0

∫ sin^6 (2x)dx =?

$$\:\int\:\mathrm{sin}\:^{\mathrm{6}} \left(\mathrm{2}{x}\right){dx}\:=?\: \\ $$

Question Number 118260 Answers: 3 Comments: 1

∫ ((x^2 −1)/(x^4 +x^2 +1)) dx

$$\int\:\frac{{x}^{\mathrm{2}} −\mathrm{1}}{{x}^{\mathrm{4}} +{x}^{\mathrm{2}} +\mathrm{1}}\:{dx} \\ $$

Question Number 118188 Answers: 1 Comments: 2

Question Number 118145 Answers: 3 Comments: 1

∫ (dx/((x+1)^2 (x^2 +1))) ?

$$\int\:\frac{{dx}}{\left({x}+\mathrm{1}\right)^{\mathrm{2}} \left({x}^{\mathrm{2}} +\mathrm{1}\right)}\:? \\ $$

Question Number 118142 Answers: 0 Comments: 0

Question Number 118113 Answers: 3 Comments: 0

∫ (dx/(x^3 (√(x^2 −a^2 )))) =?

$$\int\:\frac{{dx}}{{x}^{\mathrm{3}} \:\sqrt{{x}^{\mathrm{2}} −{a}^{\mathrm{2}} }}\:=?\: \\ $$

Question Number 118111 Answers: 2 Comments: 0

∫_0 ^(π/4) (x^2 /((x sin x+cos x)^2 )) dx =?

$$\underset{\mathrm{0}} {\overset{\pi/\mathrm{4}} {\int}}\:\frac{\mathrm{x}^{\mathrm{2}} }{\left(\mathrm{x}\:\mathrm{sin}\:\mathrm{x}+\mathrm{cos}\:\mathrm{x}\right)^{\mathrm{2}} }\:\mathrm{dx}\:=? \\ $$

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