Integration ( sin-1 (x)1/2 – cos-1 (x)1/2/sin-1 (x)1/2 + cos-1 (x)1/2) dx

Asked by haroonrashidgkp | 29th Oct, 2018, 12:05: PM

Expert Answer:

integral fraction numerator sin to the power of negative 1 end exponent left parenthesis square root of x right parenthesis end root minus space cos to the power of negative 1 end exponent left parenthesis square root of x over denominator sin to the power of negative 1 end exponent left parenthesis square root of x right parenthesis end root plus cos to the power of negative 1 end exponent left parenthesis square root of x end fraction d x

sin to the power of negative 1 end exponent left parenthesis square root of x right parenthesis end root plus cos to the power of negative 1 end exponent left parenthesis square root of x equals pi over 2
l e t space sin to the power of negative 1 end exponent left parenthesis square root of x right parenthesis end root space equals t
square root of x equals sin t space t

x equals sin squared t
d x equals 2 sin t space cos t space d t
d x equals sin space 2 t space d t

cos to the power of negative 1 end exponent x equals pi over 2 minus sin to the power of negative 1 end exponent x
h e n c e space cos to the power of negative 1 end exponent left parenthesis square root of x equals fraction numerator begin display style pi end style over denominator begin display style 2 end style end fraction minus sin to the power of negative 1 end exponent left parenthesis square root of x right parenthesis end root

integral fraction numerator begin display style sin to the power of negative 1 end exponent left parenthesis square root of x right parenthesis end root minus space cos to the power of negative 1 end exponent left parenthesis square root of x end style over denominator begin display style sin to the power of negative 1 end exponent left parenthesis square root of x right parenthesis end root plus cos to the power of negative 1 end exponent left parenthesis square root of x end style end fraction d x equals integral fraction numerator begin display style sin to the power of negative 1 end exponent left parenthesis square root of x right parenthesis end root minus space fraction numerator begin display style pi end style over denominator begin display style 2 end style end fraction plus sin to the power of negative 1 end exponent left parenthesis square root of x right parenthesis end root end style over denominator begin display style pi over 2 end style end fraction d x equals 2 over pi integral 2 sin to the power of negative 1 end exponent left parenthesis square root of x right parenthesis end root minus fraction numerator begin display style pi end style over denominator begin display style 2 end style end fraction d x
equals fraction numerator begin display style 2 end style over denominator begin display style pi end style end fraction integral left parenthesis 2 t minus fraction numerator begin display style pi end style over denominator begin display style 2 end style end fraction right parenthesis space sin space 2 t space d t
equals fraction numerator begin display style 2 end style over denominator begin display style pi end style end fraction left square bracket integral t space sin 2 t space d t minus fraction numerator begin display style pi end style over denominator begin display style 2 end style end fraction integral space sin space 2 t space d t

integral u v d x equals u integral v d x minus integral left square bracket fraction numerator d u over denominator d x end fraction integral v d x right square bracket d x
fraction numerator begin display style 2 end style over denominator begin display style pi end style end fraction left square bracket integral t space sin 2 t space d t minus fraction numerator begin display style pi end style over denominator begin display style 2 end style end fraction integral space sin space 2 t space d t equals fraction numerator begin display style 2 end style over denominator begin display style pi end style end fraction left square bracket negative t fraction numerator begin display style cos 2 t end style over denominator 2 end fraction plus integral left square bracket fraction numerator begin display style cos 2 t end style over denominator 2 end fraction right square bracket d t minus fraction numerator begin display style pi end style over denominator begin display style 2 end style end fraction fraction numerator begin display style cos 2 t end style over denominator 2 end fraction
fraction numerator begin display style 2 end style over denominator begin display style pi end style end fraction left square bracket negative t fraction numerator begin display style cos 2 t end style over denominator 2 end fraction plus fraction numerator begin display style sin space 2 t end style over denominator 4 end fraction right square bracket minus pi fraction numerator begin display style cos 2 t end style over denominator 4 end fraction
minus t fraction numerator begin display style cos 2 t end style over denominator pi end fraction plus fraction numerator begin display style sin space 2 t end style over denominator 2 pi end fraction minus fraction numerator begin display style cos 2 t end style over denominator 2 end fraction

Answered by Ram Singh | 30th Oct, 2018, 10:30: PM