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IntegrationQuestion and Answers: Page 26

Question Number 186637    Answers: 2   Comments: 0

Question Number 186531    Answers: 0   Comments: 0

Q.use the parseval relation of hankel transfrom to evaluate the Integral ∫_0 ^∞ ((J_(𝛄+1) (ar)J_(𝛄+1) (br))/r) , for 𝛄>−(1/2) , 0<a<b where J_n (x) are bessel funtions.

$$ \\ $$$$\:\:\mathbb{Q}.\boldsymbol{{use}}\:\boldsymbol{{the}}\:\boldsymbol{{parseval}}\:\boldsymbol{{relation}}\:\boldsymbol{{of}}\:\boldsymbol{{hankel}}\:\boldsymbol{{transfrom}}\:\boldsymbol{{to}}\:\boldsymbol{{evaluate}}\:\boldsymbol{{the}}\:\boldsymbol{{Integral}}\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\overset{\infty} {\int}_{\mathrm{0}} \:\:\frac{\boldsymbol{{J}}_{\boldsymbol{\gamma}+\mathrm{1}} \left(\boldsymbol{{ar}}\right)\boldsymbol{{J}}_{\boldsymbol{\gamma}+\mathrm{1}} \left(\boldsymbol{{br}}\right)}{\boldsymbol{{r}}}\:,\:\:\boldsymbol{{for}}\:\boldsymbol{\gamma}>−\frac{\mathrm{1}}{\mathrm{2}}\:,\:\:\mathrm{0}<\boldsymbol{{a}}<\boldsymbol{{b}} \\ $$$$\:\:\:\boldsymbol{{where}}\:\boldsymbol{{J}}_{\boldsymbol{{n}}} \left(\boldsymbol{{x}}\right)\:\boldsymbol{{are}}\:\boldsymbol{{bessel}}\:\boldsymbol{{funtions}}. \\ $$$$ \\ $$

Question Number 186527    Answers: 2   Comments: 0

Question Number 186476    Answers: 0   Comments: 0

Question Number 186347    Answers: 1   Comments: 1

∫_(−2) ^2 ((x^5 − 1 + 2)/(x^4 + x −2)) dx

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

Question Number 186346    Answers: 0   Comments: 0

if S_a =cos(a)+sin(x+a) then ∫(S_1 /S_2 )−((x+S_1 )/(x−S_3 ))=?

$${if}\:{S}_{{a}} =\mathrm{cos}\left({a}\right)+\mathrm{sin}\left({x}+{a}\right) \\ $$$${then}\:\int\frac{{S}_{\mathrm{1}} }{{S}_{\mathrm{2}} }−\frac{{x}+{S}_{\mathrm{1}} }{{x}−{S}_{\mathrm{3}} }=? \\ $$

Question Number 186321    Answers: 3   Comments: 0

Question Number 186310    Answers: 1   Comments: 0

((∫x(x^2 +5)^(1/2) dx − 3∫x(x^2 +5)^(−1/2) dx)/(∫ ((x[(x^2 +5)−3])/( (√(x^2 +5 )))) dx)) =??

$$ \\ $$$$\:\:\:\frac{\int\boldsymbol{{x}}\left(\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{5}\right)^{\mathrm{1}/\mathrm{2}} \boldsymbol{{dx}}\:−\:\mathrm{3}\int\boldsymbol{{x}}\left(\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{5}\right)^{−\mathrm{1}/\mathrm{2}} \:\boldsymbol{{dx}}}{\int\:\:\frac{\boldsymbol{{x}}\left[\left(\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{5}\right)−\mathrm{3}\right]}{\:\sqrt{\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{5}\:\:}}\:\boldsymbol{{dx}}}\:=??\:\:\:\: \\ $$$$ \\ $$

Question Number 186306    Answers: 0   Comments: 2

Evaluate ∫((ln(sin x))/(ln(tan x)+1)) dx

$$\mathrm{Evaluate}\:\int\frac{\mathrm{ln}\left(\mathrm{sin}\:{x}\right)}{\mathrm{ln}\left(\mathrm{tan}\:{x}\right)+\mathrm{1}}\:{dx} \\ $$

Question Number 186246    Answers: 1   Comments: 0

Question Number 186352    Answers: 1   Comments: 0

∫_1 ^( 2) ((tan^(−1) (x) + 2)/x^2 ) dx

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

Question Number 186214    Answers: 0   Comments: 12

if S_a =cos(a)+sin(x+a) then ∫(S_1 /S_2 )−((x+S_1 )/(x−S_3 ))dx=?

$${if}\:{S}_{{a}} =\mathrm{cos}\left({a}\right)+\mathrm{sin}\left({x}+{a}\right) \\ $$$${then}\:\int\frac{{S}_{\mathrm{1}} }{{S}_{\mathrm{2}} }−\frac{{x}+{S}_{\mathrm{1}} }{{x}−{S}_{\mathrm{3}} }{dx}=? \\ $$

Question Number 186198    Answers: 1   Comments: 0

∫^3 _2 ((x^2 − 1)/(1 + ^x^2 (√(2 ln(x))))) dx

$$ \\ $$$$\:\:\:\:\:\underset{\mathrm{2}} {\int}^{\mathrm{3}} \:\:\:\frac{\boldsymbol{{x}}^{\mathrm{2}} \:\:−\:\:\mathrm{1}}{\mathrm{1}\:\:\:+\:\:\:^{{x}^{\mathrm{2}} } \sqrt{\mathrm{2}\:\boldsymbol{{ln}}\left(\boldsymbol{{x}}\right)}}\:\:\:\boldsymbol{{dx}}\:\:\:\: \\ $$$$ \\ $$

Question Number 186196    Answers: 3   Comments: 2

∫_1 ^2 (((√(1 )) + cos (x))/( (√1) − cos (x))) dx

$$ \\ $$$$\:\:\:\:\:\:\:\underset{\mathrm{1}} {\overset{\mathrm{2}} {\int}}\:\:\frac{\sqrt{\mathrm{1}\:}\:\:+\:\:\boldsymbol{{cos}}\:\left(\boldsymbol{{x}}\right)}{\:\sqrt{\mathrm{1}}\:\:\:−\:\:\boldsymbol{{cos}}\:\left(\boldsymbol{{x}}\right)}\:\:\boldsymbol{{dx}}\:\:\:\:\:\: \\ $$$$ \\ $$

Question Number 186195    Answers: 1   Comments: 0

∫_1 ^2 ((1/2 ∙(x^2 ) )/(x (√(x^2 + 2)))) dx

$$ \\ $$$$\:\:\:\:\underset{\mathrm{1}} {\overset{\mathrm{2}} {\int}}\:\:\:\frac{\mathrm{1}/\mathrm{2}\:\centerdot\left(\boldsymbol{{x}}^{\mathrm{2}} \right)\:}{\boldsymbol{{x}}\:\sqrt{\boldsymbol{{x}}^{\mathrm{2}} \:+\:\:\mathrm{2}}}\:\:\boldsymbol{{dx}}\:\:\:\: \\ $$

Question Number 186194    Answers: 1   Comments: 2

∫_2 ^4 ((2x^2 − 1)/(1 + (√x^2 ) − 2)) dx

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

Question Number 186193    Answers: 2   Comments: 0

∫_0 ^1 ((sin (x))/(1 + cos(x))) dx

$$ \\ $$$$\:\:\:\:\:\:\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\:\:\frac{\boldsymbol{{sin}}\:\left(\boldsymbol{{x}}\right)}{\mathrm{1}\:\:+\:\:\boldsymbol{{cos}}\left(\boldsymbol{{x}}\right)}\:\:\boldsymbol{{dx}} \\ $$

Question Number 186192    Answers: 1   Comments: 0

My old problem ∫ e^(tan x) dx

$$ \\ $$$$\:\:\:\:\:\:\:\boldsymbol{{My}}\:\boldsymbol{{old}}\:\boldsymbol{{problem}} \\ $$$$\:\:\:\int\:\:\boldsymbol{{e}}^{\boldsymbol{{tan}}\:\boldsymbol{{x}}} \:\:\boldsymbol{{dx}} \\ $$

Question Number 186190    Answers: 0   Comments: 1

My old problem.. ∫_0 ^(+∞) ((tan^(−1) (1−cos(x)))/x^2 ) dx

$$ \\ $$$$\:\:\:\:\boldsymbol{{My}}\:\boldsymbol{{old}}\:\boldsymbol{{problem}}.. \\ $$$$\:\:\:\:\:\:\:\:\:\underset{\mathrm{0}} {\overset{+\infty} {\int}}\:\:\frac{\mathrm{tan}^{−\mathrm{1}} \left(\mathrm{1}−\boldsymbol{{cos}}\left(\boldsymbol{{x}}\right)\right)}{\boldsymbol{{x}}^{\mathrm{2}} }\:\:\boldsymbol{{dx}}\:\:\: \\ $$$$ \\ $$

Question Number 186181    Answers: 1   Comments: 0

∫_0 ^π (√(1+cos^2 x)) dx =?

$$\:\:\underset{\mathrm{0}} {\overset{\pi} {\int}}\:\sqrt{\mathrm{1}+\mathrm{cos}\:^{\mathrm{2}} {x}}\:{dx}\:=? \\ $$

Question Number 186171    Answers: 1   Comments: 0

[so easy] ∫ cos^2 (4x) + sin^4 (2x) dx

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left[\boldsymbol{{so}}\:\boldsymbol{{easy}}\right] \\ $$$$\:\:\:\:\:\:\:\int\:\boldsymbol{{cos}}^{\mathrm{2}} \:\left(\mathrm{4}\boldsymbol{{x}}\right)\:+\:\boldsymbol{{sin}}^{\mathrm{4}} \:\left(\mathrm{2}\boldsymbol{{x}}\right)\:\boldsymbol{{dx}}\:\:\: \\ $$$$ \\ $$

Question Number 186170    Answers: 1   Comments: 0

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

$$ \\ $$$$\:\:\:\:\:\:\underset{−\mathrm{2}} {\overset{\mathrm{2}} {\int}}\:\:\:\:\frac{\boldsymbol{{tan}}^{−\mathrm{1}} \:\left(\:\mathrm{2}\:−\:\boldsymbol{{cos}}\:\left(\boldsymbol{{x}}\right)\right)\:\:\:\:}{\mathrm{2}\:+\:\boldsymbol{{x}}^{\mathrm{2}} }\:\:\boldsymbol{{dx}} \\ $$$$ \\ $$

Question Number 186152    Answers: 1   Comments: 0

I= ∫_2 ^𝛑 ((cos^2 (x) − 1 )/(1 + sin (x) − tan (x))) dx

$$ \\ $$$$\:\:\:\boldsymbol{{I}}=\:\:\underset{\mathrm{2}} {\overset{\boldsymbol{\pi}} {\int}}\:\:\:\frac{\boldsymbol{{cos}}^{\mathrm{2}} \:\left(\boldsymbol{{x}}\right)\:−\:\mathrm{1}\:}{\mathrm{1}\:+\:\boldsymbol{{sin}}\:\left(\boldsymbol{{x}}\right)\:−\:\boldsymbol{{tan}}\:\left(\boldsymbol{{x}}\right)}\:\:\boldsymbol{{dx}}\:\:\:\: \\ $$$$ \\ $$

Question Number 186132    Answers: 1   Comments: 4

Question Number 186048    Answers: 0   Comments: 2

∫_a ^b (dx/( (√(b−x))+(√(x−a))))

$$\int_{{a}} ^{{b}} \frac{{dx}}{\:\sqrt{{b}−{x}}+\sqrt{{x}−{a}}} \\ $$

Question Number 185985    Answers: 2   Comments: 0

∫ (dx/( (√(sin x(1+cos x))))) =?

$$\:\:\:\int\:\frac{{dx}}{\:\sqrt{\mathrm{sin}\:{x}\left(\mathrm{1}+\mathrm{cos}\:{x}\right)}}\:=? \\ $$

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