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IntegrationQuestion and Answers: Page 21

Question Number 188384 Answers: 1 Comments: 0

if ∫_0 ^∞ e^(−ax) dx = (1/a) show that ∫_0 ^∞ e^(−ax) x^n dx = ((n!)/a^(n+1) )

$$\:\:\: \\ $$$$\:\:\:\mathrm{if}\:\:\:\:\int_{\mathrm{0}} ^{\infty} {e}^{−{ax}} {dx}\:\:=\:\:\frac{\mathrm{1}}{{a}} \\ $$$$\:\:\:\:\:\:\:\:{show}\:{that}\:\:\int_{\mathrm{0}} ^{\infty} {e}^{−{ax}} \:{x}^{{n}} {dx}\:\:=\:\:\frac{{n}!}{{a}^{{n}+\mathrm{1}} } \\ $$

Question Number 188381 Answers: 2 Comments: 0

Advanced calculus Find the value of the following series. Ω = Σ_(n=1) ^∞ (( (−1)^( n) ζ ( n ))/(n. 2^( n) )) = ? ζ ( z ) = Σ_( n=1) ^∞ (( 1)/(n^( z) )) ; Re ( z )>1

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{Advanced}\:\:\mathrm{calculus} \\ $$$$ \\ $$$$\:\:\:\:\:\mathrm{Find}\:\:\mathrm{the}\:\:\mathrm{value}\:\:\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{series}. \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\Omega\:=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\:\left(−\mathrm{1}\right)^{\:{n}} \:\zeta\:\left(\:{n}\:\right)}{{n}.\:\mathrm{2}^{\:{n}} }\:=\:? \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\zeta\:\left(\:{z}\:\right)\:=\:\underset{\:{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\:\mathrm{1}}{{n}^{\:{z}} \:}\:\:\:\:\:;\:\:\:\:\mathscr{R}{e}\:\left(\:{z}\:\right)>\mathrm{1} \\ $$$$ \\ $$

Question Number 188380 Answers: 1 Comments: 0

Question Number 188192 Answers: 2 Comments: 0

prove that ∫_0 ^∞ e^(−a^2 x^2 ) cos(2bx) dx = ((√π)/(2a))e^(−b^2 /a^2 )

$$\:\:\: \\ $$$$\:\:\:\:{prove}\:{that} \\ $$$$\:\:\:\:\:\int_{\mathrm{0}} ^{\infty} {e}^{−{a}^{\mathrm{2}} {x}^{\mathrm{2}} } \mathrm{cos}\left(\mathrm{2}{bx}\right)\:{dx}\:\:\:=\:\:\:\frac{\sqrt{\pi}}{\mathrm{2}{a}}{e}^{−{b}^{\mathrm{2}} /{a}^{\mathrm{2}} } \\ $$$$ \\ $$$$ \\ $$

Question Number 188164 Answers: 1 Comments: 1

Question Number 188086 Answers: 1 Comments: 0

I = ∫_0 ^∞ ((tan^(−1) (x/a))/(x(x^2 +b^2 )))dx

$$\:\:\:\: \\ $$$$\:\:\:\:\:\:\mathrm{I}\:\:\:=\:\:\:\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{tan}^{−\mathrm{1}} \left({x}/{a}\right)}{{x}\left({x}^{\mathrm{2}} +{b}^{\mathrm{2}} \right)}{dx} \\ $$

Question Number 188036 Answers: 1 Comments: 0

∫2^x e^x dx

$$\int\mathrm{2}^{{x}} {e}^{{x}} {dx} \\ $$

Question Number 188035 Answers: 1 Comments: 0

solve ∫(x^2 /((a+bx)^2 ))dx

$${solve} \\ $$$$\int\frac{{x}^{\mathrm{2}} }{\left({a}+{bx}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 188034 Answers: 1 Comments: 0

solve ∫((x^2 +3)/(x^6 (x^2 +1)))dx

$${solve} \\ $$$$\int\frac{{x}^{\mathrm{2}} +\mathrm{3}}{{x}^{\mathrm{6}} \left({x}^{\mathrm{2}} +\mathrm{1}\right)}{dx} \\ $$

Question Number 187993 Answers: 0 Comments: 0

prove that ∫_0 ^(π/2) ∫_0 ^(π/2) (((sin3x)/(sin2y)))^(1/3) dxdy=(π/(2(√3)))

$${prove}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \sqrt[{\mathrm{3}}]{\frac{{sin}\mathrm{3}{x}}{{sin}\mathrm{2}{y}}}{dxdy}=\frac{\pi}{\mathrm{2}\sqrt{\mathrm{3}}} \\ $$

Question Number 187898 Answers: 2 Comments: 0

∫ ((1−cos x)/(cos x+sin x−1)) dx=?

$$\:\:\int\:\frac{\mathrm{1}−\mathrm{cos}\:\mathrm{x}}{\mathrm{cos}\:\mathrm{x}+\mathrm{sin}\:\mathrm{x}−\mathrm{1}}\:\mathrm{dx}=? \\ $$

Question Number 187890 Answers: 1 Comments: 0

Question Number 187855 Answers: 2 Comments: 0

solve ∫((x^4 +x^2 +1)/(2(x^2 +1)))dx

$${solve} \\ $$$$\int\frac{{x}^{\mathrm{4}} +{x}^{\mathrm{2}} +\mathrm{1}}{\mathrm{2}\left({x}^{\mathrm{2}} +\mathrm{1}\right)}{dx} \\ $$

Question Number 187703 Answers: 2 Comments: 0

∫ (1/(5x^2 − 2x − 4)) dx

$$\int\:\frac{\mathrm{1}}{\mathrm{5x}^{\mathrm{2}} \:\:−\:\:\mathrm{2x}\:\:−\:\:\mathrm{4}}\:\mathrm{dx} \\ $$

Question Number 187700 Answers: 0 Comments: 1

Find minimum area of the part y=x^2 and y=kx(x^2 −k), k>0

$$\:{Find}\:{minimum}\:{area}\:{of}\:{the}\:{part} \\ $$$$\:{y}={x}^{\mathrm{2}} \:{and}\:{y}={kx}\left({x}^{\mathrm{2}} −{k}\right),\:{k}>\mathrm{0}\: \\ $$

Question Number 187602 Answers: 1 Comments: 0

∫_(1/4) ^(1/2) ((sin^(−1) ((√x))−cos^(−1) ((√x)))/(sin^(−1) ((√x))+cos^(−1) ((√x)))) dx=?

$$\:\:\:\underset{\mathrm{1}/\mathrm{4}} {\overset{\mathrm{1}/\mathrm{2}} {\int}}\:\frac{\mathrm{sin}^{−\mathrm{1}} \left(\sqrt{{x}}\right)−\mathrm{cos}^{−\mathrm{1}} \left(\sqrt{{x}}\right)}{\mathrm{sin}^{−\mathrm{1}} \left(\sqrt{{x}}\right)+\mathrm{cos}^{−\mathrm{1}} \left(\sqrt{{x}}\right)}\:{dx}=? \\ $$

Question Number 187531 Answers: 3 Comments: 0

∫_0 ^∞ x^2 e^(−x) dx=?

$$\underset{\mathrm{0}} {\overset{\infty} {\int}}{x}^{\mathrm{2}} {e}^{−{x}} {dx}=? \\ $$

Question Number 187530 Answers: 1 Comments: 0

∫(dx/( (√(a^2 +be^(cx) ))))=?

$$\int\frac{{dx}}{\:\sqrt{{a}^{\mathrm{2}} +{be}^{{cx}} }}=? \\ $$

Question Number 187529 Answers: 1 Comments: 0

∫x^2 ∙sin^(−1) (x)dx=?

$$\int{x}^{\mathrm{2}} \centerdot{sin}^{−\mathrm{1}} \left({x}\right){dx}=? \\ $$

Question Number 187528 Answers: 1 Comments: 0

∫_((9π)/2) ^((7π)/(1.5)) (dx/( (√(1−sinx))))=?

$$\underset{\frac{\mathrm{9}\pi}{\mathrm{2}}} {\overset{\frac{\mathrm{7}\pi}{\mathrm{1}.\mathrm{5}}} {\int}}\frac{{dx}}{\:\sqrt{\mathrm{1}−{sinx}}}=? \\ $$

Question Number 187526 Answers: 2 Comments: 0

∫(dx/(cscx−1))=?

$$\int\frac{{dx}}{{cscx}−\mathrm{1}}=? \\ $$

Question Number 187374 Answers: 0 Comments: 2

∫(√((t+1)/(t(k−t))))dt=?

$$\int\sqrt{\frac{{t}+\mathrm{1}}{{t}\left({k}−{t}\right)}}{dt}=? \\ $$

Question Number 187317 Answers: 2 Comments: 0

∫(dx/(x(√(1−2x))))

$$\int\frac{{dx}}{{x}\sqrt{\mathrm{1}−\mathrm{2}{x}}} \\ $$

Question Number 187251 Answers: 0 Comments: 1

∫ ((cos 9x)/(cos 4x. cos 2x)) dx=?

$$\:\:\int\:\frac{\mathrm{cos}\:\mathrm{9}{x}}{\mathrm{cos}\:\mathrm{4}{x}.\:\mathrm{cos}\:\mathrm{2}{x}}\:{dx}=? \\ $$

Question Number 187164 Answers: 1 Comments: 0

Question Number 187135 Answers: 1 Comments: 0

∫_( 0) ^( π/4) ((tan^2 x)/(1+sin x)) dx =?

$$\:\underset{\:\:\mathrm{0}} {\overset{\:\:\pi/\mathrm{4}} {\int}}\:\frac{\mathrm{tan}\:^{\mathrm{2}} {x}}{\mathrm{1}+\mathrm{sin}\:{x}}\:{dx}\:=? \\ $$

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