Evaluate $int fracx^2+x^2 csc ^2 xsin ^2 x$

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Evaluate $$I=int frac(x^2+x^2 csc ^2 x) :dxsin ^2 x$$



My try:



we have $$I=int x^2 csc ^2 x,dx+int x^2 csc ^4 x,dx=P+Q$$



where $$P=int x^2 csc ^2 x,dx$$ and $$Q=int x^2 csc ^4 x,dx=int x^2 csc^2 x times csc ^2 x,dx$$



Using Parts for $P$ we get



$$P=x^2 (-cot x)+int 2x cot x,dx tag1$$



Using Parts for $Q$ we get



$$Q=x^2 csc ^2 x(-cot x)+int left(-2x^2 csc^2 xcot x+2xcsc ^2 xright)cot x,dx$$ $implies$



$$Q=x^2 csc ^2 x(-cot x)+int left(-2x^2 csc^2 xcot^2 xright)dx+int 2x csc^2 xcot x,dx$$ $implies$



$$Q=x^2 csc ^2 x(-cot x)+int left(-2x^2 csc^4 xright)dx+int 2x^2 csc^2 x,dx+int 2x csc^2 xcot x,dx$$ $implies$



$$3Q=x^2 csc ^2 x(-cot x)+2P+int 2x csc^2 xcot x,dx tag2$$ Now



$$int 2x csc^2 xcot x,dx=2x cot x(-cot x)+int left (-x csc^2 x+cot x right)cot x,dx$$



$implies$



$$int 2x csc^2 xcot x,dx=frac2x cot x(-cot x)+int left (cot x right)cot x,dx3$$



$implies$



$$int 2x csc^2 xcot x,dx=frac2x cot x(-cot x)-cot x-x3$$



$$3Q-2P=x^2 csc ^2 x(-cot x)+frac2x cot x(-cot x)-cot x-x3$$



But with this can we find $P+Q$?




real-analysis integration indefinite-integrals




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  • 1




    Do you want that one checks your derivations?
    – Math-fun
    Aug 3 at 14:25










  • If you want to make sure your answer is correct, you can just go to wolframalpha.com.
    – Batominovski
    Aug 3 at 14:29










  • No i want to actually find $P+Q$ but i got $3Q-2P$
    – Ekaveera Kumar Sharma
    Aug 3 at 14:30










  • Sorry... It was a long, not-to-easy-to-read post, I didn't fully realize what you got and what you didn't.
    – Batominovski
    Aug 3 at 14:32











  • I checked with WolframAlpha, and want to warn you that the result involves a function like the polylogarithmic function $textLi_2$. And frankly, it is very nasty.
    – Batominovski
    Aug 3 at 14:35















up vote
1
down vote

favorite
3












Evaluate $$I=int frac(x^2+x^2 csc ^2 x) :dxsin ^2 x$$



My try:



we have $$I=int x^2 csc ^2 x,dx+int x^2 csc ^4 x,dx=P+Q$$



where $$P=int x^2 csc ^2 x,dx$$ and $$Q=int x^2 csc ^4 x,dx=int x^2 csc^2 x times csc ^2 x,dx$$



Using Parts for $P$ we get



$$P=x^2 (-cot x)+int 2x cot x,dx tag1$$



Using Parts for $Q$ we get



$$Q=x^2 csc ^2 x(-cot x)+int left(-2x^2 csc^2 xcot x+2xcsc ^2 xright)cot x,dx$$ $implies$



$$Q=x^2 csc ^2 x(-cot x)+int left(-2x^2 csc^2 xcot^2 xright)dx+int 2x csc^2 xcot x,dx$$ $implies$



$$Q=x^2 csc ^2 x(-cot x)+int left(-2x^2 csc^4 xright)dx+int 2x^2 csc^2 x,dx+int 2x csc^2 xcot x,dx$$ $implies$



$$3Q=x^2 csc ^2 x(-cot x)+2P+int 2x csc^2 xcot x,dx tag2$$ Now



$$int 2x csc^2 xcot x,dx=2x cot x(-cot x)+int left (-x csc^2 x+cot x right)cot x,dx$$



$implies$



$$int 2x csc^2 xcot x,dx=frac2x cot x(-cot x)+int left (cot x right)cot x,dx3$$



$implies$



$$int 2x csc^2 xcot x,dx=frac2x cot x(-cot x)-cot x-x3$$



$$3Q-2P=x^2 csc ^2 x(-cot x)+frac2x cot x(-cot x)-cot x-x3$$



But with this can we find $P+Q$?




real-analysis integration indefinite-integrals




share|cite|improve this question

















  • 1




    Do you want that one checks your derivations?
    – Math-fun
    Aug 3 at 14:25










  • If you want to make sure your answer is correct, you can just go to wolframalpha.com.
    – Batominovski
    Aug 3 at 14:29










  • No i want to actually find $P+Q$ but i got $3Q-2P$
    – Ekaveera Kumar Sharma
    Aug 3 at 14:30










  • Sorry... It was a long, not-to-easy-to-read post, I didn't fully realize what you got and what you didn't.
    – Batominovski
    Aug 3 at 14:32











  • I checked with WolframAlpha, and want to warn you that the result involves a function like the polylogarithmic function $textLi_2$. And frankly, it is very nasty.
    – Batominovski
    Aug 3 at 14:35













up vote
1
down vote

favorite
3









up vote
1
down vote

favorite
3






3





Evaluate $$I=int frac(x^2+x^2 csc ^2 x) :dxsin ^2 x$$



My try:



we have $$I=int x^2 csc ^2 x,dx+int x^2 csc ^4 x,dx=P+Q$$



where $$P=int x^2 csc ^2 x,dx$$ and $$Q=int x^2 csc ^4 x,dx=int x^2 csc^2 x times csc ^2 x,dx$$



Using Parts for $P$ we get



$$P=x^2 (-cot x)+int 2x cot x,dx tag1$$



Using Parts for $Q$ we get



$$Q=x^2 csc ^2 x(-cot x)+int left(-2x^2 csc^2 xcot x+2xcsc ^2 xright)cot x,dx$$ $implies$



$$Q=x^2 csc ^2 x(-cot x)+int left(-2x^2 csc^2 xcot^2 xright)dx+int 2x csc^2 xcot x,dx$$ $implies$



$$Q=x^2 csc ^2 x(-cot x)+int left(-2x^2 csc^4 xright)dx+int 2x^2 csc^2 x,dx+int 2x csc^2 xcot x,dx$$ $implies$



$$3Q=x^2 csc ^2 x(-cot x)+2P+int 2x csc^2 xcot x,dx tag2$$ Now



$$int 2x csc^2 xcot x,dx=2x cot x(-cot x)+int left (-x csc^2 x+cot x right)cot x,dx$$



$implies$



$$int 2x csc^2 xcot x,dx=frac2x cot x(-cot x)+int left (cot x right)cot x,dx3$$



$implies$



$$int 2x csc^2 xcot x,dx=frac2x cot x(-cot x)-cot x-x3$$



$$3Q-2P=x^2 csc ^2 x(-cot x)+frac2x cot x(-cot x)-cot x-x3$$



But with this can we find $P+Q$?




real-analysis integration indefinite-integrals




share|cite|improve this question













Evaluate $$I=int frac(x^2+x^2 csc ^2 x) :dxsin ^2 x$$



My try:



we have $$I=int x^2 csc ^2 x,dx+int x^2 csc ^4 x,dx=P+Q$$



where $$P=int x^2 csc ^2 x,dx$$ and $$Q=int x^2 csc ^4 x,dx=int x^2 csc^2 x times csc ^2 x,dx$$



Using Parts for $P$ we get



$$P=x^2 (-cot x)+int 2x cot x,dx tag1$$



Using Parts for $Q$ we get



$$Q=x^2 csc ^2 x(-cot x)+int left(-2x^2 csc^2 xcot x+2xcsc ^2 xright)cot x,dx$$ $implies$



$$Q=x^2 csc ^2 x(-cot x)+int left(-2x^2 csc^2 xcot^2 xright)dx+int 2x csc^2 xcot x,dx$$ $implies$



$$Q=x^2 csc ^2 x(-cot x)+int left(-2x^2 csc^4 xright)dx+int 2x^2 csc^2 x,dx+int 2x csc^2 xcot x,dx$$ $implies$



$$3Q=x^2 csc ^2 x(-cot x)+2P+int 2x csc^2 xcot x,dx tag2$$ Now



$$int 2x csc^2 xcot x,dx=2x cot x(-cot x)+int left (-x csc^2 x+cot x right)cot x,dx$$



$implies$



$$int 2x csc^2 xcot x,dx=frac2x cot x(-cot x)+int left (cot x right)cot x,dx3$$



$implies$



$$int 2x csc^2 xcot x,dx=frac2x cot x(-cot x)-cot x-x3$$



$$3Q-2P=x^2 csc ^2 x(-cot x)+frac2x cot x(-cot x)-cot x-x3$$



But with this can we find $P+Q$?





real-analysis integration indefinite-integrals





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Aug 3 at 17:22









NickD

9081412




9081412









asked Aug 3 at 14:21









Ekaveera Kumar Sharma

5,13311122




5,13311122







  • 1




    Do you want that one checks your derivations?
    – Math-fun
    Aug 3 at 14:25










  • If you want to make sure your answer is correct, you can just go to wolframalpha.com.
    – Batominovski
    Aug 3 at 14:29










  • No i want to actually find $P+Q$ but i got $3Q-2P$
    – Ekaveera Kumar Sharma
    Aug 3 at 14:30










  • Sorry... It was a long, not-to-easy-to-read post, I didn't fully realize what you got and what you didn't.
    – Batominovski
    Aug 3 at 14:32











  • I checked with WolframAlpha, and want to warn you that the result involves a function like the polylogarithmic function $textLi_2$. And frankly, it is very nasty.
    – Batominovski
    Aug 3 at 14:35













  • 1




    Do you want that one checks your derivations?
    – Math-fun
    Aug 3 at 14:25










  • If you want to make sure your answer is correct, you can just go to wolframalpha.com.
    – Batominovski
    Aug 3 at 14:29










  • No i want to actually find $P+Q$ but i got $3Q-2P$
    – Ekaveera Kumar Sharma
    Aug 3 at 14:30










  • Sorry... It was a long, not-to-easy-to-read post, I didn't fully realize what you got and what you didn't.
    – Batominovski
    Aug 3 at 14:32











  • I checked with WolframAlpha, and want to warn you that the result involves a function like the polylogarithmic function $textLi_2$. And frankly, it is very nasty.
    – Batominovski
    Aug 3 at 14:35








1




1




Do you want that one checks your derivations?
– Math-fun
Aug 3 at 14:25




Do you want that one checks your derivations?
– Math-fun
Aug 3 at 14:25












If you want to make sure your answer is correct, you can just go to wolframalpha.com.
– Batominovski
Aug 3 at 14:29




If you want to make sure your answer is correct, you can just go to wolframalpha.com.
– Batominovski
Aug 3 at 14:29












No i want to actually find $P+Q$ but i got $3Q-2P$
– Ekaveera Kumar Sharma
Aug 3 at 14:30




No i want to actually find $P+Q$ but i got $3Q-2P$
– Ekaveera Kumar Sharma
Aug 3 at 14:30












Sorry... It was a long, not-to-easy-to-read post, I didn't fully realize what you got and what you didn't.
– Batominovski
Aug 3 at 14:32





Sorry... It was a long, not-to-easy-to-read post, I didn't fully realize what you got and what you didn't.
– Batominovski
Aug 3 at 14:32













I checked with WolframAlpha, and want to warn you that the result involves a function like the polylogarithmic function $textLi_2$. And frankly, it is very nasty.
– Batominovski
Aug 3 at 14:35





I checked with WolframAlpha, and want to warn you that the result involves a function like the polylogarithmic function $textLi_2$. And frankly, it is very nasty.
– Batominovski
Aug 3 at 14:35
















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