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IntegrationQuestion and Answers: Page 101

Question Number 134866 Answers: 2 Comments: 0

let U_n =∫_0 ^∞ ((cos(nx))/((x^2 +n^2 )^2 ))dx calculate lim_(n→+∞) e^n^2 U_n

$$\mathrm{let}\:\mathrm{U}_{\mathrm{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{cos}\left(\mathrm{nx}\right)}{\left(\mathrm{x}^{\mathrm{2}} \:+\mathrm{n}^{\mathrm{2}} \right)^{\mathrm{2}} }\mathrm{dx}\:\mathrm{calculate}\:\:\mathrm{lim}_{\mathrm{n}\rightarrow+\infty} \mathrm{e}^{\mathrm{n}^{\mathrm{2}} } \mathrm{U}_{\mathrm{n}} \\ $$

Question Number 134845 Answers: 1 Comments: 0

advanced calculus... ∫_0 ^( 1) xψ(1+x)dx=?? ...m.n...

$$\:\:\:\:\:\:\:\:\:\:\:{advanced}\:\:{calculus}... \\ $$$$\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} {x}\psi\left(\mathrm{1}+{x}\right){dx}=?? \\ $$$$\:\:\:\:\:\:\:\:...{m}.{n}... \\ $$

Question Number 134841 Answers: 0 Comments: 0

∫_0 ^1 y^2 ln(((y^2 −1)/y^2 ))ln(((y+1)/y))dy From Q134820

$$\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{y}^{\mathrm{2}} \mathrm{ln}\left(\frac{\mathrm{y}^{\mathrm{2}} −\mathrm{1}}{\mathrm{y}^{\mathrm{2}} }\right)\mathrm{ln}\left(\frac{\mathrm{y}+\mathrm{1}}{\mathrm{y}}\right)\mathrm{dy} \\ $$$${From}\:{Q}\mathrm{134820} \\ $$

Question Number 134820 Answers: 1 Comments: 0

Question Number 134811 Answers: 1 Comments: 0

∫_0 ^( π/2) ((sin (((3x)/2)))/(tan (3x))) dx

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

Question Number 134795 Answers: 1 Comments: 0

Question Number 134764 Answers: 1 Comments: 0

Z = ∫_0 ^( 2) (dx/( (√(∣x−1∣)))) ?

$$\:\mathbb{Z}\:=\:\int_{\mathrm{0}} ^{\:\mathrm{2}} \:\frac{\mathrm{dx}}{\:\sqrt{\mid\mathrm{x}−\mathrm{1}\mid}}\:? \\ $$

Question Number 134828 Answers: 0 Comments: 0

∫^( (π/2)) _0 ((sin (((2x)/3)))/(tan (x))) dx =?

$$\underset{\mathrm{0}} {\int}^{\:\frac{\pi}{\mathrm{2}}} \frac{\mathrm{sin}\:\left(\frac{\mathrm{2}{x}}{\mathrm{3}}\right)}{\mathrm{tan}\:\left({x}\right)}\:{dx}\:=?\: \\ $$

Question Number 134750 Answers: 0 Comments: 0

.....advanced calculus.... prove that:: 𝛗=Σ(1/(n^3 sin((√2) πn))) =((−13(√2) π^3 )/(720)) ...m.n...

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:.....{advanced}\:\:\:{calculus}.... \\ $$$$\:\:\:{prove}\:\:\:{that}::\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\phi}=\Sigma\frac{\mathrm{1}}{{n}^{\mathrm{3}} {sin}\left(\sqrt{\mathrm{2}}\:\pi{n}\right)}\:=\frac{−\mathrm{13}\sqrt{\mathrm{2}}\:\pi^{\mathrm{3}} \:}{\mathrm{720}} \\ $$$$\:\:\:\:\:...{m}.{n}... \\ $$

Question Number 134716 Answers: 1 Comments: 0

∫((ln(x))/(x−1))dx=...??

$$\int\frac{{ln}\left({x}\right)}{{x}−\mathrm{1}}{dx}=...?? \\ $$

Question Number 134833 Answers: 1 Comments: 0

∫ ((x^3 −1)/(4x^3 −x)) dx ?

$$\:\int\:\frac{{x}^{\mathrm{3}} −\mathrm{1}}{\mathrm{4}{x}^{\mathrm{3}} −{x}}\:{dx}\:? \\ $$

Question Number 134708 Answers: 2 Comments: 0

D = ∫_0 ^( π/2) sin^4 x cos^5 x dx

$$\mathscr{D}\:=\:\int_{\mathrm{0}} ^{\:\pi/\mathrm{2}} \mathrm{sin}\:^{\mathrm{4}} \mathrm{x}\:\mathrm{cos}\:^{\mathrm{5}} \mathrm{x}\:\mathrm{dx}\: \\ $$

Question Number 134693 Answers: 1 Comments: 0

∫sec^4 xtanxdx??

$$\int{sec}^{\mathrm{4}} {xtanxdx}?? \\ $$

Question Number 134672 Answers: 0 Comments: 0

....nice calculus... prove that : (√(5π))≤ ∫_0 ^( (π/3)) ((9(√3))/(2π))(√(8sin(x)−sin(2x))) dx≤(√(6π)) ...m.n....

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:....{nice}\:\:\:\:{calculus}... \\ $$$$\:\:\:\:\:{prove}\:\:{that}\:: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\sqrt{\mathrm{5}\pi}\leqslant\:\:\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{3}}} \frac{\mathrm{9}\sqrt{\mathrm{3}}}{\mathrm{2}\pi}\sqrt{\mathrm{8}{sin}\left({x}\right)−{sin}\left(\mathrm{2}{x}\right)}\:{dx}\leqslant\sqrt{\mathrm{6}\pi} \\ $$$$\:\:\:\:\:\:\:\:...{m}.{n}.... \\ $$

Question Number 134667 Answers: 0 Comments: 1

G = ∫_0 ^( π/2) arccos (((cos x)/(1+2cos x))) dx

$$\mathcal{G}\:=\:\int_{\mathrm{0}} ^{\:\pi/\mathrm{2}} \:\mathrm{arccos}\:\left(\frac{\mathrm{cos}\:\mathrm{x}}{\mathrm{1}+\mathrm{2cos}\:\mathrm{x}}\right)\:\mathrm{dx}\: \\ $$

Question Number 134660 Answers: 2 Comments: 0

M = ∫ (dx/(2cos x+3sin x))

$$\mathscr{M}\:=\:\int\:\frac{{dx}}{\mathrm{2cos}\:{x}+\mathrm{3sin}\:{x}}\: \\ $$

Question Number 134641 Answers: 4 Comments: 0

B = ∫ ((1−sin 6x)/(1+sin 6x)) dx

$$\mathscr{B}\:=\:\int\:\frac{\mathrm{1}−\mathrm{sin}\:\mathrm{6}{x}}{\mathrm{1}+\mathrm{sin}\:\mathrm{6}{x}}\:{dx}\: \\ $$

Question Number 134636 Answers: 4 Comments: 0

INTEGRAL (1)∫_0 ^( ln 2) x^(−2) .e^(−(1/x)) dx =? (2) ∫ ((sin x+cos x)/(sin^4 x+cos^4 x)) dx =?

$$\mathcal{INTEGRAL} \\ $$$$\left(\mathrm{1}\right)\int_{\mathrm{0}} ^{\:\mathrm{ln}\:\mathrm{2}} \:{x}^{−\mathrm{2}} .{e}^{−\frac{\mathrm{1}}{{x}}} \:{dx}\:=? \\ $$$$\left(\mathrm{2}\right)\:\int\:\frac{\mathrm{sin}\:\mathrm{x}+\mathrm{cos}\:\mathrm{x}}{\mathrm{sin}\:^{\mathrm{4}} \mathrm{x}+\mathrm{cos}\:^{\mathrm{4}} \mathrm{x}}\:\mathrm{dx}\:=? \\ $$

Question Number 134619 Answers: 0 Comments: 0

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

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

Question Number 134601 Answers: 2 Comments: 0

∫3((√(x + (4/( (√x)))))) dx

$$\int\mathrm{3}\left(\sqrt{\mathrm{x}\:+\:\frac{\mathrm{4}}{\:\sqrt{\mathrm{x}}}}\right)\:\mathrm{dx} \\ $$

Question Number 134591 Answers: 1 Comments: 0

Precalculus What is the volume of the solid obtained by rotating the region bounded by x = (y−2)^2 and y = x about the line y = −1 by the method of cylindrical shells?

$$\mathrm{Precalculus} \\ $$What is the volume of the solid obtained by rotating the region bounded by x = (y−2)^2 and y = x about the line y = −1 by the method of cylindrical shells?

Question Number 134587 Answers: 0 Comments: 2

Precalculus How do I find the volume of a solid obtained by rotating the region bounded by x=(y−3)^2 and x=4 about y=1?

$$\mathrm{Precalculus}\: \\ $$How do I find the volume of a solid obtained by rotating the region bounded by x=(y−3)^2 and x=4 about y=1?

Question Number 134583 Answers: 0 Comments: 0

Question Number 134580 Answers: 2 Comments: 0

∫(√(tan(x)))dx=...?

$$\int\sqrt{{tan}\left({x}\right)}{dx}=...? \\ $$

Question Number 134570 Answers: 0 Comments: 0

∫_0 ^( 1) ∫_y ^( y^(1/3) ) (√(x^4 +1)) dx dy =?

$$\int_{\mathrm{0}} ^{\:\mathrm{1}} \int_{\mathrm{y}} ^{\:\mathrm{y}^{\mathrm{1}/\mathrm{3}} } \sqrt{\mathrm{x}^{\mathrm{4}} +\mathrm{1}}\:\mathrm{dx}\:\mathrm{dy}\:=?\:\: \\ $$

Question Number 134547 Answers: 2 Comments: 0

R = ∫_0 ^( 3a) x (√((3a−x)/(x+a))) dx

$$\mathscr{R}\:=\:\int_{\mathrm{0}} ^{\:\mathrm{3}{a}} {x}\:\sqrt{\frac{\mathrm{3}{a}−{x}}{{x}+{a}}}\:{dx}\: \\ $$

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