What will be the integration of :
1/(1+under root of tan x) 

Asked by nikhil | 15th Oct, 2015, 09:43: PM

Expert Answer:

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 d x over denominator 1 plus square root of t a n x end root end fraction N o w space s u b s t i t u t i n g comma space t a n x equals t squared semicolon space D i f f e r e n t i a t i n g space w e space space h a v e comma space s e c squared x space d x equals 2 t space d t T h u s comma space I equals integral fraction numerator 2 t space d t over denominator s e c squared x open parentheses 1 plus t close parentheses end fraction space space equals integral fraction numerator 2 t space d t over denominator open parentheses 1 plus t a n squared x close parentheses open parentheses 1 plus t close parentheses end fraction space space equals integral fraction numerator 2 t space d t over denominator open parentheses 1 plus t to the power of 4 close parentheses open parentheses 1 plus t close parentheses end fraction N o w space r e w r i t i n g space t h e space i n t e g r a n d comma space w e space h a v e comma fraction numerator 2 t space over denominator open parentheses 1 plus t to the power of 4 close parentheses open parentheses 1 plus t close parentheses end fraction equals fraction numerator A t cubed plus B t squared plus C t plus D over denominator open parentheses 1 plus t to the power of 4 close parentheses end fraction plus fraction numerator E over denominator 1 plus t end fraction T h u s comma space w e space s o l v i n g space t h e space a b o v e space p a r t i a l space f r a c t i o n s comma space w e space h a v e comma fraction numerator 2 t space over denominator open parentheses 1 plus t to the power of 4 close parentheses open parentheses 1 plus t close parentheses end fraction equals fraction numerator t cubed minus t squared plus t plus 1 over denominator open parentheses 1 plus t to the power of 4 close parentheses end fraction minus fraction numerator 1 over denominator 1 plus t end fraction  A g a i n space c o n s i d e r space t h e space f r a c t i o n comma space fraction numerator t cubed minus t squared plus t plus 1 over denominator open parentheses 1 plus t to the power of 4 close parentheses end fraction equals fraction numerator t cubed minus t squared plus t plus 1 over denominator open parentheses t squared plus square root of 2 t plus 1 close parentheses open parentheses t squared minus square root of 2 t plus 1 close parentheses end fraction A p p l y i n g space p a r t i a l space f r a c t i o n s space a g a i n comma space w e space h a v e comma fraction numerator t cubed minus t squared plus t plus 1 over denominator open parentheses 1 plus t to the power of 4 close parentheses end fraction equals fraction numerator P t plus Q over denominator open parentheses t squared plus square root of 2 t plus 1 close parentheses end fraction plus fraction numerator R t plus S over denominator open parentheses t squared minus square root of 2 t plus 1 close parentheses end fraction S o l v i n g space t h e space a b o v e space p a r t i a l space f r a c t i o n s comma space w e space h a v e comma fraction numerator t cubed minus t squared plus t plus 1 over denominator open parentheses 1 plus t to the power of 4 close parentheses end fraction equals fraction numerator begin display style fraction numerator square root of 2 plus 2 over denominator 2 square root of 2 end fraction t plus 1 half end style over denominator open parentheses t squared plus square root of 2 t plus 1 close parentheses end fraction plus fraction numerator fraction numerator square root of 2 minus 2 over denominator 2 square root of 2 end fraction t plus 1 half over denominator open parentheses t squared minus square root of 2 t plus 1 close parentheses end fraction T h u s comma space t h e space g i v e n space i n t e g r a l space b e c o m e s comma I equals integral open square brackets fraction numerator begin display style fraction numerator square root of 2 plus 2 over denominator 2 square root of 2 end fraction t plus 1 half end style over denominator open parentheses t squared plus square root of 2 t plus 1 close parentheses end fraction plus fraction numerator fraction numerator square root of 2 minus 2 over denominator 2 square root of 2 end fraction t plus 1 half over denominator open parentheses t squared minus square root of 2 t plus 1 close parentheses end fraction minus fraction numerator 1 over denominator 1 plus t end fraction close square brackets d t rightwards double arrow I equals integral open square brackets fraction numerator begin display style fraction numerator square root of 2 plus 2 over denominator 2 square root of 2 end fraction t plus 1 half end style over denominator open parentheses t squared plus square root of 2 t plus 1 close parentheses end fraction close square brackets d t plus integral open square brackets fraction numerator fraction numerator square root of 2 minus 2 over denominator 2 square root of 2 end fraction t plus 1 half over denominator open parentheses t squared minus square root of 2 t plus 1 close parentheses end fraction close square brackets d t minus integral open square brackets fraction numerator 1 over denominator 1 plus t end fraction close square brackets d t rightwards double arrow I equals fraction numerator square root of 2 plus 2 over denominator 4 square root of 2 end fraction integral open square brackets fraction numerator begin display style 2 t plus square root of 2 end style over denominator open parentheses t squared plus square root of 2 t plus 1 close parentheses end fraction close square brackets d t plus integral open square brackets fraction numerator begin display style 1 half minus fraction numerator 2 plus square root of 2 over denominator 4 end fraction end style over denominator open parentheses t squared plus square root of 2 t plus 1 close parentheses end fraction close square brackets d t plus fraction numerator square root of 2 minus 2 over denominator 4 square root of 2 end fraction integral open square brackets fraction numerator 2 t minus square root of 2 over denominator open parentheses t squared minus square root of 2 t plus 1 close parentheses end fraction close square brackets d t plus integral open square brackets fraction numerator 1 half plus fraction numerator square root of 2 minus 2 over denominator 4 end fraction over denominator open parentheses t squared minus square root of 2 t plus 1 close parentheses end fraction close square brackets d t minus integral open square brackets fraction numerator 1 over denominator 1 plus t end fraction close square brackets d t rightwards double arrow I equals fraction numerator square root of 2 plus 2 over denominator 4 square root of 2 end fraction l n open parentheses t squared plus square root of 2 t plus 1 close parentheses plus open square brackets 1 half minus fraction numerator 2 plus square root of 2 over denominator 4 end fraction close square brackets cross times square root of 2 t a n to the power of negative 1 end exponent open parentheses square root of 2 t plus 1 close parentheses plus fraction numerator square root of 2 minus 2 over denominator 4 square root of 2 end fraction l n open parentheses t squared minus square root of 2 t plus 1 close parentheses plus open square brackets 1 half plus fraction numerator square root of 2 minus 2 over denominator 4 end fraction close square brackets square root of 2 t a n to the power of negative 1 end exponent open parentheses square root of 2 t minus 1 close parentheses minus l n open parentheses 1 plus t close parentheses plus C w h e r e space t equals square root of t a n x end root space a n d space C equals c o n s t a n t

Answered by Vimala Ramamurthy | 16th Oct, 2015, 11:58: AM