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IntegrationQuestion and Answers: Page 26
Question Number 186637 Answers: 2 Comments: 0
Question Number 186531 Answers: 0 Comments: 0
$$ \\ $$$$\:\:\mathbb{Q}.\boldsymbol{{use}}\:\boldsymbol{{the}}\:\boldsymbol{{parseval}}\:\boldsymbol{{relation}}\:\boldsymbol{{of}}\:\boldsymbol{{hankel}}\:\boldsymbol{{transfrom}}\:\boldsymbol{{to}}\:\boldsymbol{{evaluate}}\:\boldsymbol{{the}}\:\boldsymbol{{Integral}}\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\overset{\infty} {\int}_{\mathrm{0}} \:\:\frac{\boldsymbol{{J}}_{\boldsymbol{\gamma}+\mathrm{1}} \left(\boldsymbol{{ar}}\right)\boldsymbol{{J}}_{\boldsymbol{\gamma}+\mathrm{1}} \left(\boldsymbol{{br}}\right)}{\boldsymbol{{r}}}\:,\:\:\boldsymbol{{for}}\:\boldsymbol{\gamma}>−\frac{\mathrm{1}}{\mathrm{2}}\:,\:\:\mathrm{0}<\boldsymbol{{a}}<\boldsymbol{{b}} \\ $$$$\:\:\:\boldsymbol{{where}}\:\boldsymbol{{J}}_{\boldsymbol{{n}}} \left(\boldsymbol{{x}}\right)\:\boldsymbol{{are}}\:\boldsymbol{{bessel}}\:\boldsymbol{{funtions}}. \\ $$$$ \\ $$
Question Number 186527 Answers: 2 Comments: 0
Question Number 186476 Answers: 0 Comments: 0
Question Number 186347 Answers: 1 Comments: 1
$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\underset{−\mathrm{2}} {\overset{\mathrm{2}} {\int}}\:\:\:\:\frac{\boldsymbol{{x}}^{\mathrm{5}} \:−\:\:\mathrm{1}\:\:+\:\:\mathrm{2}}{\boldsymbol{{x}}^{\mathrm{4}} \:\:+\:\:\boldsymbol{{x}}\:\:−\mathrm{2}}\:\:\:\:\boldsymbol{{dx}}\:\:\:\:\:\:\: \\ $$$$ \\ $$
Question Number 186346 Answers: 0 Comments: 0
$${if}\:{S}_{{a}} =\mathrm{cos}\left({a}\right)+\mathrm{sin}\left({x}+{a}\right) \\ $$$${then}\:\int\frac{{S}_{\mathrm{1}} }{{S}_{\mathrm{2}} }−\frac{{x}+{S}_{\mathrm{1}} }{{x}−{S}_{\mathrm{3}} }=? \\ $$
Question Number 186321 Answers: 3 Comments: 0
Question Number 186310 Answers: 1 Comments: 0
$$ \\ $$$$\:\:\:\frac{\int\boldsymbol{{x}}\left(\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{5}\right)^{\mathrm{1}/\mathrm{2}} \boldsymbol{{dx}}\:−\:\mathrm{3}\int\boldsymbol{{x}}\left(\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{5}\right)^{−\mathrm{1}/\mathrm{2}} \:\boldsymbol{{dx}}}{\int\:\:\frac{\boldsymbol{{x}}\left[\left(\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{5}\right)−\mathrm{3}\right]}{\:\sqrt{\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{5}\:\:}}\:\boldsymbol{{dx}}}\:=??\:\:\:\: \\ $$$$ \\ $$
Question Number 186306 Answers: 0 Comments: 2
$$\mathrm{Evaluate}\:\int\frac{\mathrm{ln}\left(\mathrm{sin}\:{x}\right)}{\mathrm{ln}\left(\mathrm{tan}\:{x}\right)+\mathrm{1}}\:{dx} \\ $$
Question Number 186246 Answers: 1 Comments: 0
Question Number 186352 Answers: 1 Comments: 0
$$ \\ $$$$\:\:\:\:\:\:\:\:\:\int_{\mathrm{1}} ^{\:\mathrm{2}} \:\:\frac{\boldsymbol{{tan}}^{−\mathrm{1}} \:\left(\boldsymbol{{x}}\right)\:+\:\mathrm{2}}{\boldsymbol{{x}}^{\mathrm{2}} }\:\:\boldsymbol{{dx}}\:\:\:\:\:\:\: \\ $$$$ \\ $$
Question Number 186214 Answers: 0 Comments: 12
$${if}\:{S}_{{a}} =\mathrm{cos}\left({a}\right)+\mathrm{sin}\left({x}+{a}\right) \\ $$$${then}\:\int\frac{{S}_{\mathrm{1}} }{{S}_{\mathrm{2}} }−\frac{{x}+{S}_{\mathrm{1}} }{{x}−{S}_{\mathrm{3}} }{dx}=? \\ $$
Question Number 186198 Answers: 1 Comments: 0
$$ \\ $$$$\:\:\:\:\:\underset{\mathrm{2}} {\int}^{\mathrm{3}} \:\:\:\frac{\boldsymbol{{x}}^{\mathrm{2}} \:\:−\:\:\mathrm{1}}{\mathrm{1}\:\:\:+\:\:\:^{{x}^{\mathrm{2}} } \sqrt{\mathrm{2}\:\boldsymbol{{ln}}\left(\boldsymbol{{x}}\right)}}\:\:\:\boldsymbol{{dx}}\:\:\:\: \\ $$$$ \\ $$
Question Number 186196 Answers: 3 Comments: 2
$$ \\ $$$$\:\:\:\:\:\:\:\underset{\mathrm{1}} {\overset{\mathrm{2}} {\int}}\:\:\frac{\sqrt{\mathrm{1}\:}\:\:+\:\:\boldsymbol{{cos}}\:\left(\boldsymbol{{x}}\right)}{\:\sqrt{\mathrm{1}}\:\:\:−\:\:\boldsymbol{{cos}}\:\left(\boldsymbol{{x}}\right)}\:\:\boldsymbol{{dx}}\:\:\:\:\:\: \\ $$$$ \\ $$
Question Number 186195 Answers: 1 Comments: 0
$$ \\ $$$$\:\:\:\:\underset{\mathrm{1}} {\overset{\mathrm{2}} {\int}}\:\:\:\frac{\mathrm{1}/\mathrm{2}\:\centerdot\left(\boldsymbol{{x}}^{\mathrm{2}} \right)\:}{\boldsymbol{{x}}\:\sqrt{\boldsymbol{{x}}^{\mathrm{2}} \:+\:\:\mathrm{2}}}\:\:\boldsymbol{{dx}}\:\:\:\: \\ $$
Question Number 186194 Answers: 1 Comments: 2
$$ \\ $$$$\:\:\:\:\:\:\underset{\mathrm{2}} {\overset{\mathrm{4}} {\int}}\:\:\frac{\mathrm{2}\boldsymbol{{x}}^{\mathrm{2}} \:−\:\mathrm{1}}{\mathrm{1}\:+\:\:\sqrt{\boldsymbol{{x}}^{\mathrm{2}} }\:\:−\:\:\mathrm{2}}\:\:\boldsymbol{{dx}}\:\:\:\: \\ $$$$ \\ $$
Question Number 186193 Answers: 2 Comments: 0
$$ \\ $$$$\:\:\:\:\:\:\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\:\:\frac{\boldsymbol{{sin}}\:\left(\boldsymbol{{x}}\right)}{\mathrm{1}\:\:+\:\:\boldsymbol{{cos}}\left(\boldsymbol{{x}}\right)}\:\:\boldsymbol{{dx}} \\ $$
Question Number 186192 Answers: 1 Comments: 0
$$ \\ $$$$\:\:\:\:\:\:\:\boldsymbol{{My}}\:\boldsymbol{{old}}\:\boldsymbol{{problem}} \\ $$$$\:\:\:\int\:\:\boldsymbol{{e}}^{\boldsymbol{{tan}}\:\boldsymbol{{x}}} \:\:\boldsymbol{{dx}} \\ $$
Question Number 186190 Answers: 0 Comments: 1
$$ \\ $$$$\:\:\:\:\boldsymbol{{My}}\:\boldsymbol{{old}}\:\boldsymbol{{problem}}.. \\ $$$$\:\:\:\:\:\:\:\:\:\underset{\mathrm{0}} {\overset{+\infty} {\int}}\:\:\frac{\mathrm{tan}^{−\mathrm{1}} \left(\mathrm{1}−\boldsymbol{{cos}}\left(\boldsymbol{{x}}\right)\right)}{\boldsymbol{{x}}^{\mathrm{2}} }\:\:\boldsymbol{{dx}}\:\:\: \\ $$$$ \\ $$
Question Number 186181 Answers: 1 Comments: 0
$$\:\:\underset{\mathrm{0}} {\overset{\pi} {\int}}\:\sqrt{\mathrm{1}+\mathrm{cos}\:^{\mathrm{2}} {x}}\:{dx}\:=? \\ $$
Question Number 186171 Answers: 1 Comments: 0
$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left[\boldsymbol{{so}}\:\boldsymbol{{easy}}\right] \\ $$$$\:\:\:\:\:\:\:\int\:\boldsymbol{{cos}}^{\mathrm{2}} \:\left(\mathrm{4}\boldsymbol{{x}}\right)\:+\:\boldsymbol{{sin}}^{\mathrm{4}} \:\left(\mathrm{2}\boldsymbol{{x}}\right)\:\boldsymbol{{dx}}\:\:\: \\ $$$$ \\ $$
Question Number 186170 Answers: 1 Comments: 0
$$ \\ $$$$\:\:\:\:\:\:\underset{−\mathrm{2}} {\overset{\mathrm{2}} {\int}}\:\:\:\:\frac{\boldsymbol{{tan}}^{−\mathrm{1}} \:\left(\:\mathrm{2}\:−\:\boldsymbol{{cos}}\:\left(\boldsymbol{{x}}\right)\right)\:\:\:\:}{\mathrm{2}\:+\:\boldsymbol{{x}}^{\mathrm{2}} }\:\:\boldsymbol{{dx}} \\ $$$$ \\ $$
Question Number 186152 Answers: 1 Comments: 0
$$ \\ $$$$\:\:\:\boldsymbol{{I}}=\:\:\underset{\mathrm{2}} {\overset{\boldsymbol{\pi}} {\int}}\:\:\:\frac{\boldsymbol{{cos}}^{\mathrm{2}} \:\left(\boldsymbol{{x}}\right)\:−\:\mathrm{1}\:}{\mathrm{1}\:+\:\boldsymbol{{sin}}\:\left(\boldsymbol{{x}}\right)\:−\:\boldsymbol{{tan}}\:\left(\boldsymbol{{x}}\right)}\:\:\boldsymbol{{dx}}\:\:\:\: \\ $$$$ \\ $$
Question Number 186132 Answers: 1 Comments: 4
Question Number 186048 Answers: 0 Comments: 2
$$\int_{{a}} ^{{b}} \frac{{dx}}{\:\sqrt{{b}−{x}}+\sqrt{{x}−{a}}} \\ $$
Question Number 185985 Answers: 2 Comments: 0
$$\:\:\:\int\:\frac{{dx}}{\:\sqrt{\mathrm{sin}\:{x}\left(\mathrm{1}+\mathrm{cos}\:{x}\right)}}\:=? \\ $$
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