`int (x^(2)dx)/(sqrt(4+x^(6)))` का मान ज्ञात कीजिये ।

Explanation

माना `I = int (x^(2) dx)/(sqrt((4 +x^(6)))) = (1)/(2) int (x^(2)dx)/(sqrt(1+((x^(3))/(2))^(2)))`
माना `(x^(3))/(2) = tan theta rArr (3x^(2))/(2) dx = sec^(2) theta d theta`
`rArr" "x^(2)dx = (2)/(3)sec^(2) theta d theta`
`therefore" "I = (1)/(2)xx(2)/(3)int(sec^(2) theta d theta)/(sqrt(1+ tan^(2) theta)) = (1)/(3) intsec theta d theta`
`= (1)/(3)log(sec theta + tan theta)`
पुन : `(x^(3))/(2) = tan theta`
चित्र से `sec theta = (sqrt(x^(6) +4))/(2)`
तथा ` tan theta =(x^(2))/(2)`
`therefore" "I = (1)/(3)log[(x^(3) + sqrt(x^(6) +4))/(2)]`
` = (1)/(3)[log[x^(3) + sqrt((x^(6)+4)]]-log_(2)]`
`log_(2)`को समाकलन स्थिराक में सम्मिलित करने पर
`= (1)/(3)log[x^(3) + sqrt((x^(6) + 4))]`