Question
Sun March 15, 2015 By: Jerrin Thomas

integrate x/(1+x^3)

Expert Reply
Vimala Ramamurthy
Mon March 16, 2015
C o n s i d e r space t h e space i n t e g r a l space I equals integral fraction numerator x over denominator 1 plus x cubed end fraction d x W e space k n o w space t h a t space a cubed plus b cubed equals open parentheses a plus b close parentheses open parentheses a squared minus a b plus b squared close parentheses T h u s comma space 1 plus x cubed equals open parentheses 1 plus x close parentheses open parentheses x squared minus x plus 1 close parentheses space I equals integral fraction numerator x over denominator 1 plus x cubed end fraction d x equals integral fraction numerator x over denominator open parentheses 1 plus x close parentheses open parentheses x squared minus x plus 1 close parentheses end fraction d x L e t space u s space u s e space p a r t i a l space f r a c t i o o n s space t o space s e p a r a t e space t h e space f r a c t i o n s. T h u s comma fraction numerator x over denominator open parentheses 1 plus x close parentheses open parentheses x squared minus x plus 1 close parentheses end fraction equals fraction numerator A over denominator 1 plus x end fraction plus fraction numerator B x plus C over denominator x squared minus x plus 1 end fraction rightwards double arrow x equals A open parentheses x squared minus x plus 1 close parentheses plus open parentheses B x plus C close parentheses open parentheses 1 plus x close parentheses rightwards double arrow x equals A x squared minus A x plus A plus B x plus B x squared plus C plus C x C o m p a r i n g space t h e space c o e f f i c i e n t s comma space w e space h a v e comma A plus B equals 0 semicolon minus A plus B plus C equals 1 semicolon A plus C equals 0 S o l v i n g space t h e space a b o v e space s y s t m e space o f space e q u a t i o n s comma space w e space h a v e comma A equals negative 1 third comma space B equals 1 third space a n d space C equals 1 third T h u s comma space w e space h a v e comma fraction numerator x over denominator open parentheses 1 plus x close parentheses open parentheses x squared minus x plus 1 close parentheses end fraction equals fraction numerator negative 1 third over denominator 1 plus x end fraction plus fraction numerator 1 third x plus 1 third over denominator x squared minus x plus 1 end fraction T h e r e f o r e comma space I equals integral open square brackets fraction numerator negative 1 third over denominator 1 plus x end fraction plus fraction numerator 1 third x plus 1 third over denominator x squared minus x plus 1 end fraction close square brackets d x equals negative 1 third integral open square brackets fraction numerator 1 over denominator 1 plus x end fraction close square brackets d x plus 1 third integral open square brackets fraction numerator x plus 1 over denominator x squared minus x plus 1 end fraction close square brackets d x N o w space c o n s i d e r space t h e space i n t e g r a l space I subscript 1 equals 1 third integral open square brackets fraction numerator x plus 1 over denominator x squared minus x plus 1 end fraction close square brackets d x L e t space u s space u s e space i n t e g r a t i o n space b y space p a r t s u equals x plus 1 semicolon d u equals d x integral fraction numerator 1 over denominator x squared minus x plus 1 end fraction d x equals integral d v rightwards double arrow integral fraction numerator 1 over denominator x squared minus x plus begin display style 1 fourth end style plus 1 minus 1 fourth end fraction d x equals integral d v rightwards double arrow integral fraction numerator 1 over denominator open parentheses x minus begin display style 1 half end style close parentheses squared plus 3 over 4 end fraction d x equals integral d v rightwards double arrow integral fraction numerator 1 over denominator open parentheses x minus begin display style 1 half end style close parentheses squared plus open parentheses fraction numerator square root of 3 over denominator 2 end fraction close parentheses squared end fraction d x equals integral d v rightwards double arrow fraction numerator 1 over denominator fraction numerator square root of 3 over denominator 2 end fraction end fraction tan to the power of negative 1 end exponent fraction numerator open parentheses x minus 1 half close parentheses over denominator fraction numerator square root of 3 over denominator 2 end fraction end fraction equals v rightwards double arrow fraction numerator 2 over denominator square root of 3 end fraction tan to the power of negative 1 end exponent fraction numerator open parentheses 2 x minus 1 close parentheses over denominator square root of 3 end fraction equals v T h u s comma I subscript 1 equals u v minus integral v d u rightwards double arrow 1 third open square brackets fraction numerator 2 open parentheses x plus 1 close parentheses over denominator square root of 3 end fraction tan to the power of negative 1 end exponent fraction numerator open parentheses 2 x minus 1 close parentheses over denominator square root of 3 end fraction minus integral fraction numerator 2 over denominator square root of 3 end fraction tan to the power of negative 1 end exponent fraction numerator open parentheses 2 x minus 1 close parentheses over denominator square root of 3 end fraction d x close square brackets rightwards double arrow 1 third open square brackets fraction numerator 2 open parentheses x plus 1 close parentheses over denominator square root of 3 end fraction tan to the power of negative 1 end exponent fraction numerator open parentheses 2 x minus 1 close parentheses over denominator square root of 3 end fraction minus fraction numerator 2 over denominator square root of 3 end fraction integral tan to the power of negative 1 end exponent fraction numerator open parentheses 2 x minus 1 close parentheses over denominator square root of 3 end fraction d x close square brackets C o n s i d e r space t h e space i n t e g r a t i o n integral tan to the power of negative 1 end exponent fraction numerator open parentheses 2 x minus 1 close parentheses over denominator square root of 3 end fraction d x T h u s comma 2 x minus 1 equals s rightwards double arrow 2 d x equals d s rightwards double arrow d x equals fraction numerator d s over denominator 2 end fraction integral tan to the power of negative 1 end exponent fraction numerator open parentheses 2 x minus 1 close parentheses over denominator square root of 3 end fraction d x equals 1 half integral tan to the power of negative 1 end exponent fraction numerator open parentheses s close parentheses over denominator square root of 3 end fraction d s A g a i n space l e t space u s space u s e space i n t e g r a t i o n space b y space p a r t s space t o space i n t e g r a t e space integral tan to the power of negative 1 end exponent fraction numerator open parentheses s close parentheses over denominator square root of 3 end fraction d s u equals tan to the power of negative 1 end exponent fraction numerator open parentheses s close parentheses over denominator square root of 3 end fraction rightwards double arrow d u equals fraction numerator 1 over denominator 3 plus s squared end fraction cross times fraction numerator 1 over denominator begin display style fraction numerator 1 over denominator square root of 3 end fraction end style end fraction cross times d s equals fraction numerator square root of 3 over denominator 3 plus s squared end fraction d s integral d v equals integral d s rightwards double arrow v equals s T h u s integral tan to the power of negative 1 end exponent fraction numerator open parentheses s close parentheses over denominator square root of 3 end fraction d s equals s tan to the power of negative 1 end exponent fraction numerator open parentheses s close parentheses over denominator square root of 3 end fraction minus integral fraction numerator s square root of 3 over denominator 3 plus s squared end fraction d s rightwards double arrow integral tan to the power of negative 1 end exponent fraction numerator open parentheses s close parentheses over denominator square root of 3 end fraction d s equals s tan to the power of negative 1 end exponent fraction numerator open parentheses s close parentheses over denominator square root of 3 end fraction minus square root of 3 integral fraction numerator s over denominator 3 plus s squared end fraction d s rightwards double arrow integral tan to the power of negative 1 end exponent fraction numerator open parentheses s close parentheses over denominator square root of 3 end fraction d s equals s tan to the power of negative 1 end exponent fraction numerator open parentheses s close parentheses over denominator square root of 3 end fraction minus fraction numerator square root of 3 over denominator 2 end fraction ln open parentheses 3 plus s squared close parentheses T h u s comma integral tan to the power of negative 1 end exponent fraction numerator open parentheses 2 x minus 1 close parentheses over denominator square root of 3 end fraction d x equals 1 half open square brackets open parentheses 2 x minus 1 close parentheses tan to the power of negative 1 end exponent fraction numerator open parentheses 2 x minus 1 close parentheses over denominator square root of 3 end fraction minus fraction numerator square root of 3 over denominator 2 end fraction ln open square brackets open parentheses 2 x minus 1 close parentheses squared plus 3 close square brackets close square brackets A n d space w e space h a v e comma 1 third integral open square brackets fraction numerator x plus 1 over denominator x squared minus x plus 1 end fraction close square brackets d x equals 1 third open square brackets fraction numerator 2 open parentheses x plus 1 close parentheses over denominator square root of 3 end fraction tan to the power of negative 1 end exponent fraction numerator open parentheses 2 x minus 1 close parentheses over denominator square root of 3 end fraction minus fraction numerator 2 over denominator square root of 3 end fraction 1 half open square brackets open parentheses 2 x minus 1 close parentheses tan to the power of negative 1 end exponent fraction numerator open parentheses 2 x minus 1 close parentheses over denominator square root of 3 end fraction minus fraction numerator square root of 3 over denominator 2 end fraction ln open square brackets open parentheses 2 x minus 1 close parentheses squared plus 3 close square brackets close square brackets close square brackets integral fraction numerator x over denominator 1 plus x cubed end fraction d x equals negative 1 third integral open square brackets fraction numerator 1 over denominator 1 plus x end fraction close square brackets d x plus 1 third integral open square brackets fraction numerator x plus 1 over denominator x squared minus x plus 1 end fraction close square brackets d x equals negative 1 third log open parentheses 1 plus x close parentheses plus 1 third open square brackets fraction numerator 2 open parentheses x plus 1 close parentheses over denominator square root of 3 end fraction tan to the power of negative 1 end exponent fraction numerator open parentheses 2 x minus 1 close parentheses over denominator square root of 3 end fraction minus fraction numerator 2 over denominator square root of 3 end fraction 1 half open square brackets open parentheses 2 x minus 1 close parentheses tan to the power of negative 1 end exponent fraction numerator open parentheses 2 x minus 1 close parentheses over denominator square root of 3 end fraction minus fraction numerator square root of 3 over denominator 2 end fraction ln open square brackets open parentheses 2 x minus 1 close parentheses squared plus 3 close square brackets close square brackets close square brackets plus C
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