Improper integral $int_0^infty frac{x^{alpha}ln x}{x^2+1},dx=frac{pi^2}{4} frac{sin(pi alpha/2)}{cos^2(pi...











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$$int_0^infty frac{x^{alpha}ln x}{x^2+1},dx=frac{pi^2}{4} frac{sin(pi alpha/2)}{cos^2(pi alpha/2)}$$
where $0 < alpha < 1$.




Answer: When i put this term in my integral calculator, it gave me very lengthy answer involving polylogarithm functions.










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closed as off-topic by TheSimpliFire, Holo, Did, José Carlos Santos, Brahadeesh Nov 25 at 11:12


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – TheSimpliFire, Holo, Did, José Carlos Santos, Brahadeesh

If this question can be reworded to fit the rules in the help center, please edit the question.













  • Have you searched in this site about this integral?
    – Nosrati
    Nov 19 at 3:59










  • @Nosrati, I didn't search in this site about this integral. But if this is a duplicate question, please provide the link of this question's answer available in this site.
    – Dhamnekar Winod
    Nov 19 at 4:08















up vote
-3
down vote

favorite













$$int_0^infty frac{x^{alpha}ln x}{x^2+1},dx=frac{pi^2}{4} frac{sin(pi alpha/2)}{cos^2(pi alpha/2)}$$
where $0 < alpha < 1$.




Answer: When i put this term in my integral calculator, it gave me very lengthy answer involving polylogarithm functions.










share|cite|improve this question















closed as off-topic by TheSimpliFire, Holo, Did, José Carlos Santos, Brahadeesh Nov 25 at 11:12


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – TheSimpliFire, Holo, Did, José Carlos Santos, Brahadeesh

If this question can be reworded to fit the rules in the help center, please edit the question.













  • Have you searched in this site about this integral?
    – Nosrati
    Nov 19 at 3:59










  • @Nosrati, I didn't search in this site about this integral. But if this is a duplicate question, please provide the link of this question's answer available in this site.
    – Dhamnekar Winod
    Nov 19 at 4:08













up vote
-3
down vote

favorite









up vote
-3
down vote

favorite












$$int_0^infty frac{x^{alpha}ln x}{x^2+1},dx=frac{pi^2}{4} frac{sin(pi alpha/2)}{cos^2(pi alpha/2)}$$
where $0 < alpha < 1$.




Answer: When i put this term in my integral calculator, it gave me very lengthy answer involving polylogarithm functions.










share|cite|improve this question
















$$int_0^infty frac{x^{alpha}ln x}{x^2+1},dx=frac{pi^2}{4} frac{sin(pi alpha/2)}{cos^2(pi alpha/2)}$$
where $0 < alpha < 1$.




Answer: When i put this term in my integral calculator, it gave me very lengthy answer involving polylogarithm functions.







calculus integration improper-integrals self-learning polylogarithm






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share|cite|improve this question













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edited Nov 19 at 4:25









Nosrati

26.3k62353




26.3k62353










asked Nov 19 at 3:55









Dhamnekar Winod

361414




361414




closed as off-topic by TheSimpliFire, Holo, Did, José Carlos Santos, Brahadeesh Nov 25 at 11:12


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – TheSimpliFire, Holo, Did, José Carlos Santos, Brahadeesh

If this question can be reworded to fit the rules in the help center, please edit the question.




closed as off-topic by TheSimpliFire, Holo, Did, José Carlos Santos, Brahadeesh Nov 25 at 11:12


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – TheSimpliFire, Holo, Did, José Carlos Santos, Brahadeesh

If this question can be reworded to fit the rules in the help center, please edit the question.












  • Have you searched in this site about this integral?
    – Nosrati
    Nov 19 at 3:59










  • @Nosrati, I didn't search in this site about this integral. But if this is a duplicate question, please provide the link of this question's answer available in this site.
    – Dhamnekar Winod
    Nov 19 at 4:08


















  • Have you searched in this site about this integral?
    – Nosrati
    Nov 19 at 3:59










  • @Nosrati, I didn't search in this site about this integral. But if this is a duplicate question, please provide the link of this question's answer available in this site.
    – Dhamnekar Winod
    Nov 19 at 4:08
















Have you searched in this site about this integral?
– Nosrati
Nov 19 at 3:59




Have you searched in this site about this integral?
– Nosrati
Nov 19 at 3:59












@Nosrati, I didn't search in this site about this integral. But if this is a duplicate question, please provide the link of this question's answer available in this site.
– Dhamnekar Winod
Nov 19 at 4:08




@Nosrati, I didn't search in this site about this integral. But if this is a duplicate question, please provide the link of this question's answer available in this site.
– Dhamnekar Winod
Nov 19 at 4:08










2 Answers
2






active

oldest

votes

















up vote
5
down vote



accepted










$$
begin{align}
int_0^inftyfrac{x^alphalog(x)}{x^2+1},mathrm{d}x
&=frac{mathrm{d}}{mathrm{d}alpha}int_0^inftyfrac{x^alpha}{x^2+1},mathrm{d}xtag1\
&=frac12frac{mathrm{d}}{mathrm{d}alpha}int_0^inftyfrac{x^{frac{alpha-1}2}}{x+1},mathrm{d}xtag2\
&=frac12frac{mathrm{d}}{mathrm{d}alpha}frac{Gammaleft(frac{1+alpha}2right)Gammaleft(frac{1-alpha}2right)}{Gamma(1)}tag3\
&=fracpi2frac{mathrm{d}}{mathrm{d}alpha}cscleft(pifrac{1-alpha}2right)tag4\
&=fracpi2frac{mathrm{d}}{mathrm{d}alpha}secleft(frac{pialpha}2right)tag5\
&=frac{pi^2}4tanleft(frac{pialpha}2right)secleft(frac{pialpha}2right)tag6
end{align}
$$

Explanation:
$(1)$: $frac{mathrm{d}}{mathrm{d}alpha}x^alpha=x^alphalog(x)$
$(2)$: substitute $xmapstosqrt{x}$
$(3)$: Beta Function
$(4)$: Euler Reflection Formula
$(5)$: trigonometric identity
$(6)$: evaluate the derivative






share|cite|improve this answer





















  • How did you compute (2)? You substitute x by $sqrt{x}$.So I get $frac12frac{d}{dalpha}intlimits_0^inftyfrac{x^frac{alpha}{2}}{x+1}dx$ But your term in (2) is different?
    – Dhamnekar Winod
    Nov 19 at 13:28










  • What is $mathrm{d}sqrt{x}$? You got the $frac12$, but forgot the $x^{-1/2}$.
    – robjohn
    Nov 19 at 13:58










  • I already got the answers to this question from you and one other member of this site correctly satisfying all my queries. Then what is the use of making it off-topic and put on hold?
    – Dhamnekar Winod
    Nov 25 at 12:35


















up vote
4
down vote













By substitution $x=tan t$
$$I(a)=int_0^inftydfrac{x^a}{1+x^2} dx=int_0^{pi/2}tan^at dt$$
then using dear Beta function
$$I(a)=dfrac12Gammaleft(dfrac{a+1}{2}right)Gammaleft(dfrac{-a+1}{2}right)=dfrac{pi}{2sinpidfrac{1-a}{2}}=dfrac{pi}{2cosdfrac{api}{2}}$$
by Reflection formula the desired integral is
$$dfrac{d}{da}I(a)=color{blue}{dfrac{pi^2}{4}dfrac{sinfrac{api}{2}}{cos^2frac{api}{2}}}$$






share|cite|improve this answer



















  • 1




    How did you compute the first equation? I never found it in the list of trigonometric identities?
    – Dhamnekar Winod
    Nov 19 at 4:59






  • 1




    It is in fact no trigonometric identity but just the substitution $x=tan(t)$. From hereon you will get the upper as well as the lower border of integration. Further note that the derivative of $x=tan(t)$ can be written as $dt=frac{dx}{1+x^2}$ due the derivative of the tangent function.
    – mrtaurho
    Nov 19 at 10:48










  • @Nosrati, is $I(a)$ regularised incomplete beta function? why it is multiplied by$frac12$?
    – Dhamnekar Winod
    Nov 20 at 5:35










  • See the link, fourth formula of Properties.
    – Nosrati
    Nov 20 at 5:37






  • 1




    So $$I(a)=int_0^{pi/2}tan^at dt=dfrac12left(2int_0^{pi/2}sin^atcos^{-a}t dtright)$$
    – Nosrati
    Nov 20 at 6:36


















2 Answers
2






active

oldest

votes








2 Answers
2






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
5
down vote



accepted










$$
begin{align}
int_0^inftyfrac{x^alphalog(x)}{x^2+1},mathrm{d}x
&=frac{mathrm{d}}{mathrm{d}alpha}int_0^inftyfrac{x^alpha}{x^2+1},mathrm{d}xtag1\
&=frac12frac{mathrm{d}}{mathrm{d}alpha}int_0^inftyfrac{x^{frac{alpha-1}2}}{x+1},mathrm{d}xtag2\
&=frac12frac{mathrm{d}}{mathrm{d}alpha}frac{Gammaleft(frac{1+alpha}2right)Gammaleft(frac{1-alpha}2right)}{Gamma(1)}tag3\
&=fracpi2frac{mathrm{d}}{mathrm{d}alpha}cscleft(pifrac{1-alpha}2right)tag4\
&=fracpi2frac{mathrm{d}}{mathrm{d}alpha}secleft(frac{pialpha}2right)tag5\
&=frac{pi^2}4tanleft(frac{pialpha}2right)secleft(frac{pialpha}2right)tag6
end{align}
$$

Explanation:
$(1)$: $frac{mathrm{d}}{mathrm{d}alpha}x^alpha=x^alphalog(x)$
$(2)$: substitute $xmapstosqrt{x}$
$(3)$: Beta Function
$(4)$: Euler Reflection Formula
$(5)$: trigonometric identity
$(6)$: evaluate the derivative






share|cite|improve this answer





















  • How did you compute (2)? You substitute x by $sqrt{x}$.So I get $frac12frac{d}{dalpha}intlimits_0^inftyfrac{x^frac{alpha}{2}}{x+1}dx$ But your term in (2) is different?
    – Dhamnekar Winod
    Nov 19 at 13:28










  • What is $mathrm{d}sqrt{x}$? You got the $frac12$, but forgot the $x^{-1/2}$.
    – robjohn
    Nov 19 at 13:58










  • I already got the answers to this question from you and one other member of this site correctly satisfying all my queries. Then what is the use of making it off-topic and put on hold?
    – Dhamnekar Winod
    Nov 25 at 12:35















up vote
5
down vote



accepted










$$
begin{align}
int_0^inftyfrac{x^alphalog(x)}{x^2+1},mathrm{d}x
&=frac{mathrm{d}}{mathrm{d}alpha}int_0^inftyfrac{x^alpha}{x^2+1},mathrm{d}xtag1\
&=frac12frac{mathrm{d}}{mathrm{d}alpha}int_0^inftyfrac{x^{frac{alpha-1}2}}{x+1},mathrm{d}xtag2\
&=frac12frac{mathrm{d}}{mathrm{d}alpha}frac{Gammaleft(frac{1+alpha}2right)Gammaleft(frac{1-alpha}2right)}{Gamma(1)}tag3\
&=fracpi2frac{mathrm{d}}{mathrm{d}alpha}cscleft(pifrac{1-alpha}2right)tag4\
&=fracpi2frac{mathrm{d}}{mathrm{d}alpha}secleft(frac{pialpha}2right)tag5\
&=frac{pi^2}4tanleft(frac{pialpha}2right)secleft(frac{pialpha}2right)tag6
end{align}
$$

Explanation:
$(1)$: $frac{mathrm{d}}{mathrm{d}alpha}x^alpha=x^alphalog(x)$
$(2)$: substitute $xmapstosqrt{x}$
$(3)$: Beta Function
$(4)$: Euler Reflection Formula
$(5)$: trigonometric identity
$(6)$: evaluate the derivative






share|cite|improve this answer





















  • How did you compute (2)? You substitute x by $sqrt{x}$.So I get $frac12frac{d}{dalpha}intlimits_0^inftyfrac{x^frac{alpha}{2}}{x+1}dx$ But your term in (2) is different?
    – Dhamnekar Winod
    Nov 19 at 13:28










  • What is $mathrm{d}sqrt{x}$? You got the $frac12$, but forgot the $x^{-1/2}$.
    – robjohn
    Nov 19 at 13:58










  • I already got the answers to this question from you and one other member of this site correctly satisfying all my queries. Then what is the use of making it off-topic and put on hold?
    – Dhamnekar Winod
    Nov 25 at 12:35













up vote
5
down vote



accepted







up vote
5
down vote



accepted






$$
begin{align}
int_0^inftyfrac{x^alphalog(x)}{x^2+1},mathrm{d}x
&=frac{mathrm{d}}{mathrm{d}alpha}int_0^inftyfrac{x^alpha}{x^2+1},mathrm{d}xtag1\
&=frac12frac{mathrm{d}}{mathrm{d}alpha}int_0^inftyfrac{x^{frac{alpha-1}2}}{x+1},mathrm{d}xtag2\
&=frac12frac{mathrm{d}}{mathrm{d}alpha}frac{Gammaleft(frac{1+alpha}2right)Gammaleft(frac{1-alpha}2right)}{Gamma(1)}tag3\
&=fracpi2frac{mathrm{d}}{mathrm{d}alpha}cscleft(pifrac{1-alpha}2right)tag4\
&=fracpi2frac{mathrm{d}}{mathrm{d}alpha}secleft(frac{pialpha}2right)tag5\
&=frac{pi^2}4tanleft(frac{pialpha}2right)secleft(frac{pialpha}2right)tag6
end{align}
$$

Explanation:
$(1)$: $frac{mathrm{d}}{mathrm{d}alpha}x^alpha=x^alphalog(x)$
$(2)$: substitute $xmapstosqrt{x}$
$(3)$: Beta Function
$(4)$: Euler Reflection Formula
$(5)$: trigonometric identity
$(6)$: evaluate the derivative






share|cite|improve this answer












$$
begin{align}
int_0^inftyfrac{x^alphalog(x)}{x^2+1},mathrm{d}x
&=frac{mathrm{d}}{mathrm{d}alpha}int_0^inftyfrac{x^alpha}{x^2+1},mathrm{d}xtag1\
&=frac12frac{mathrm{d}}{mathrm{d}alpha}int_0^inftyfrac{x^{frac{alpha-1}2}}{x+1},mathrm{d}xtag2\
&=frac12frac{mathrm{d}}{mathrm{d}alpha}frac{Gammaleft(frac{1+alpha}2right)Gammaleft(frac{1-alpha}2right)}{Gamma(1)}tag3\
&=fracpi2frac{mathrm{d}}{mathrm{d}alpha}cscleft(pifrac{1-alpha}2right)tag4\
&=fracpi2frac{mathrm{d}}{mathrm{d}alpha}secleft(frac{pialpha}2right)tag5\
&=frac{pi^2}4tanleft(frac{pialpha}2right)secleft(frac{pialpha}2right)tag6
end{align}
$$

Explanation:
$(1)$: $frac{mathrm{d}}{mathrm{d}alpha}x^alpha=x^alphalog(x)$
$(2)$: substitute $xmapstosqrt{x}$
$(3)$: Beta Function
$(4)$: Euler Reflection Formula
$(5)$: trigonometric identity
$(6)$: evaluate the derivative







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Nov 19 at 7:25









robjohn

263k27301623




263k27301623












  • How did you compute (2)? You substitute x by $sqrt{x}$.So I get $frac12frac{d}{dalpha}intlimits_0^inftyfrac{x^frac{alpha}{2}}{x+1}dx$ But your term in (2) is different?
    – Dhamnekar Winod
    Nov 19 at 13:28










  • What is $mathrm{d}sqrt{x}$? You got the $frac12$, but forgot the $x^{-1/2}$.
    – robjohn
    Nov 19 at 13:58










  • I already got the answers to this question from you and one other member of this site correctly satisfying all my queries. Then what is the use of making it off-topic and put on hold?
    – Dhamnekar Winod
    Nov 25 at 12:35


















  • How did you compute (2)? You substitute x by $sqrt{x}$.So I get $frac12frac{d}{dalpha}intlimits_0^inftyfrac{x^frac{alpha}{2}}{x+1}dx$ But your term in (2) is different?
    – Dhamnekar Winod
    Nov 19 at 13:28










  • What is $mathrm{d}sqrt{x}$? You got the $frac12$, but forgot the $x^{-1/2}$.
    – robjohn
    Nov 19 at 13:58










  • I already got the answers to this question from you and one other member of this site correctly satisfying all my queries. Then what is the use of making it off-topic and put on hold?
    – Dhamnekar Winod
    Nov 25 at 12:35
















How did you compute (2)? You substitute x by $sqrt{x}$.So I get $frac12frac{d}{dalpha}intlimits_0^inftyfrac{x^frac{alpha}{2}}{x+1}dx$ But your term in (2) is different?
– Dhamnekar Winod
Nov 19 at 13:28




How did you compute (2)? You substitute x by $sqrt{x}$.So I get $frac12frac{d}{dalpha}intlimits_0^inftyfrac{x^frac{alpha}{2}}{x+1}dx$ But your term in (2) is different?
– Dhamnekar Winod
Nov 19 at 13:28












What is $mathrm{d}sqrt{x}$? You got the $frac12$, but forgot the $x^{-1/2}$.
– robjohn
Nov 19 at 13:58




What is $mathrm{d}sqrt{x}$? You got the $frac12$, but forgot the $x^{-1/2}$.
– robjohn
Nov 19 at 13:58












I already got the answers to this question from you and one other member of this site correctly satisfying all my queries. Then what is the use of making it off-topic and put on hold?
– Dhamnekar Winod
Nov 25 at 12:35




I already got the answers to this question from you and one other member of this site correctly satisfying all my queries. Then what is the use of making it off-topic and put on hold?
– Dhamnekar Winod
Nov 25 at 12:35










up vote
4
down vote













By substitution $x=tan t$
$$I(a)=int_0^inftydfrac{x^a}{1+x^2} dx=int_0^{pi/2}tan^at dt$$
then using dear Beta function
$$I(a)=dfrac12Gammaleft(dfrac{a+1}{2}right)Gammaleft(dfrac{-a+1}{2}right)=dfrac{pi}{2sinpidfrac{1-a}{2}}=dfrac{pi}{2cosdfrac{api}{2}}$$
by Reflection formula the desired integral is
$$dfrac{d}{da}I(a)=color{blue}{dfrac{pi^2}{4}dfrac{sinfrac{api}{2}}{cos^2frac{api}{2}}}$$






share|cite|improve this answer



















  • 1




    How did you compute the first equation? I never found it in the list of trigonometric identities?
    – Dhamnekar Winod
    Nov 19 at 4:59






  • 1




    It is in fact no trigonometric identity but just the substitution $x=tan(t)$. From hereon you will get the upper as well as the lower border of integration. Further note that the derivative of $x=tan(t)$ can be written as $dt=frac{dx}{1+x^2}$ due the derivative of the tangent function.
    – mrtaurho
    Nov 19 at 10:48










  • @Nosrati, is $I(a)$ regularised incomplete beta function? why it is multiplied by$frac12$?
    – Dhamnekar Winod
    Nov 20 at 5:35










  • See the link, fourth formula of Properties.
    – Nosrati
    Nov 20 at 5:37






  • 1




    So $$I(a)=int_0^{pi/2}tan^at dt=dfrac12left(2int_0^{pi/2}sin^atcos^{-a}t dtright)$$
    – Nosrati
    Nov 20 at 6:36















up vote
4
down vote













By substitution $x=tan t$
$$I(a)=int_0^inftydfrac{x^a}{1+x^2} dx=int_0^{pi/2}tan^at dt$$
then using dear Beta function
$$I(a)=dfrac12Gammaleft(dfrac{a+1}{2}right)Gammaleft(dfrac{-a+1}{2}right)=dfrac{pi}{2sinpidfrac{1-a}{2}}=dfrac{pi}{2cosdfrac{api}{2}}$$
by Reflection formula the desired integral is
$$dfrac{d}{da}I(a)=color{blue}{dfrac{pi^2}{4}dfrac{sinfrac{api}{2}}{cos^2frac{api}{2}}}$$






share|cite|improve this answer



















  • 1




    How did you compute the first equation? I never found it in the list of trigonometric identities?
    – Dhamnekar Winod
    Nov 19 at 4:59






  • 1




    It is in fact no trigonometric identity but just the substitution $x=tan(t)$. From hereon you will get the upper as well as the lower border of integration. Further note that the derivative of $x=tan(t)$ can be written as $dt=frac{dx}{1+x^2}$ due the derivative of the tangent function.
    – mrtaurho
    Nov 19 at 10:48










  • @Nosrati, is $I(a)$ regularised incomplete beta function? why it is multiplied by$frac12$?
    – Dhamnekar Winod
    Nov 20 at 5:35










  • See the link, fourth formula of Properties.
    – Nosrati
    Nov 20 at 5:37






  • 1




    So $$I(a)=int_0^{pi/2}tan^at dt=dfrac12left(2int_0^{pi/2}sin^atcos^{-a}t dtright)$$
    – Nosrati
    Nov 20 at 6:36













up vote
4
down vote










up vote
4
down vote









By substitution $x=tan t$
$$I(a)=int_0^inftydfrac{x^a}{1+x^2} dx=int_0^{pi/2}tan^at dt$$
then using dear Beta function
$$I(a)=dfrac12Gammaleft(dfrac{a+1}{2}right)Gammaleft(dfrac{-a+1}{2}right)=dfrac{pi}{2sinpidfrac{1-a}{2}}=dfrac{pi}{2cosdfrac{api}{2}}$$
by Reflection formula the desired integral is
$$dfrac{d}{da}I(a)=color{blue}{dfrac{pi^2}{4}dfrac{sinfrac{api}{2}}{cos^2frac{api}{2}}}$$






share|cite|improve this answer














By substitution $x=tan t$
$$I(a)=int_0^inftydfrac{x^a}{1+x^2} dx=int_0^{pi/2}tan^at dt$$
then using dear Beta function
$$I(a)=dfrac12Gammaleft(dfrac{a+1}{2}right)Gammaleft(dfrac{-a+1}{2}right)=dfrac{pi}{2sinpidfrac{1-a}{2}}=dfrac{pi}{2cosdfrac{api}{2}}$$
by Reflection formula the desired integral is
$$dfrac{d}{da}I(a)=color{blue}{dfrac{pi^2}{4}dfrac{sinfrac{api}{2}}{cos^2frac{api}{2}}}$$







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Nov 19 at 5:33

























answered Nov 19 at 4:15









Nosrati

26.3k62353




26.3k62353








  • 1




    How did you compute the first equation? I never found it in the list of trigonometric identities?
    – Dhamnekar Winod
    Nov 19 at 4:59






  • 1




    It is in fact no trigonometric identity but just the substitution $x=tan(t)$. From hereon you will get the upper as well as the lower border of integration. Further note that the derivative of $x=tan(t)$ can be written as $dt=frac{dx}{1+x^2}$ due the derivative of the tangent function.
    – mrtaurho
    Nov 19 at 10:48










  • @Nosrati, is $I(a)$ regularised incomplete beta function? why it is multiplied by$frac12$?
    – Dhamnekar Winod
    Nov 20 at 5:35










  • See the link, fourth formula of Properties.
    – Nosrati
    Nov 20 at 5:37






  • 1




    So $$I(a)=int_0^{pi/2}tan^at dt=dfrac12left(2int_0^{pi/2}sin^atcos^{-a}t dtright)$$
    – Nosrati
    Nov 20 at 6:36














  • 1




    How did you compute the first equation? I never found it in the list of trigonometric identities?
    – Dhamnekar Winod
    Nov 19 at 4:59






  • 1




    It is in fact no trigonometric identity but just the substitution $x=tan(t)$. From hereon you will get the upper as well as the lower border of integration. Further note that the derivative of $x=tan(t)$ can be written as $dt=frac{dx}{1+x^2}$ due the derivative of the tangent function.
    – mrtaurho
    Nov 19 at 10:48










  • @Nosrati, is $I(a)$ regularised incomplete beta function? why it is multiplied by$frac12$?
    – Dhamnekar Winod
    Nov 20 at 5:35










  • See the link, fourth formula of Properties.
    – Nosrati
    Nov 20 at 5:37






  • 1




    So $$I(a)=int_0^{pi/2}tan^at dt=dfrac12left(2int_0^{pi/2}sin^atcos^{-a}t dtright)$$
    – Nosrati
    Nov 20 at 6:36








1




1




How did you compute the first equation? I never found it in the list of trigonometric identities?
– Dhamnekar Winod
Nov 19 at 4:59




How did you compute the first equation? I never found it in the list of trigonometric identities?
– Dhamnekar Winod
Nov 19 at 4:59




1




1




It is in fact no trigonometric identity but just the substitution $x=tan(t)$. From hereon you will get the upper as well as the lower border of integration. Further note that the derivative of $x=tan(t)$ can be written as $dt=frac{dx}{1+x^2}$ due the derivative of the tangent function.
– mrtaurho
Nov 19 at 10:48




It is in fact no trigonometric identity but just the substitution $x=tan(t)$. From hereon you will get the upper as well as the lower border of integration. Further note that the derivative of $x=tan(t)$ can be written as $dt=frac{dx}{1+x^2}$ due the derivative of the tangent function.
– mrtaurho
Nov 19 at 10:48












@Nosrati, is $I(a)$ regularised incomplete beta function? why it is multiplied by$frac12$?
– Dhamnekar Winod
Nov 20 at 5:35




@Nosrati, is $I(a)$ regularised incomplete beta function? why it is multiplied by$frac12$?
– Dhamnekar Winod
Nov 20 at 5:35












See the link, fourth formula of Properties.
– Nosrati
Nov 20 at 5:37




See the link, fourth formula of Properties.
– Nosrati
Nov 20 at 5:37




1




1




So $$I(a)=int_0^{pi/2}tan^at dt=dfrac12left(2int_0^{pi/2}sin^atcos^{-a}t dtright)$$
– Nosrati
Nov 20 at 6:36




So $$I(a)=int_0^{pi/2}tan^at dt=dfrac12left(2int_0^{pi/2}sin^atcos^{-a}t dtright)$$
– Nosrati
Nov 20 at 6:36



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