Find the integral of the function w. r. to x.

Asked by Topperlearning User | 30th Oct, 2016, 02:55: PM

Expert Answer:

integral square root of x squared plus a squared end root d x
equals p u t space x equals a tan z rightwards double arrow d x equals a s e c squared z d z
t h e space i n t e g r a l space b e c o m e s
J equals integral square root of a squared tan squared z plus a squared end root a s e c squared z d z
J equals integral a s e c z a s e c squared z d z
J equals a squared integral s e c cubed z d z
equals a squared integral s e c squared z s e c z d z
equals a squared open parentheses s e c z integral s e c squared z d z minus integral open parentheses fraction numerator d over denominator d z end fraction s e c z integral s e c squared z d z close parentheses d z close parentheses
equals a squared open parentheses s e c z tan z minus integral s e c z tan squared z d z close parentheses
equals a squared open parentheses s e c z tan z minus integral s e c z left parenthesis s e c squared z minus 1 right parenthesis d z close parentheses
equals a squared s e c z tan z minus a squared integral s e c cubed z d z plus a squared integral s e c z d z
equals a squared s e c z tan z minus J plus a squared log open vertical bar s e c z plus tan z close vertical bar plus c o n s tan t
2 J equals a squared s e c z tan z plus a squared log open vertical bar s e c z plus tan z close vertical bar plus c o n s tan t
J equals 1 half open parentheses a squared s e c z tan z plus a squared log open vertical bar s e c z plus tan z close vertical bar close parentheses plus c o n s tan t
space space space equals a squared over 2 open parentheses x over a square root of 1 plus x squared over a squared end root plus log open vertical bar square root of 1 plus x squared over a squared end root plus x over a close vertical bar close parentheses plus c o n s tan t
space space space equals open parentheses x over 2 square root of x squared plus a squared end root plus a squared over 2 log open vertical bar square root of x squared plus a squared end root plus x close vertical bar close parentheses plus c o n s tan t

Answered by  | 30th Oct, 2016, 04:55: PM