The discussion centers on the correctness of a solution involving trigonometric substitution for the integral I(1/(9x^2+6x-8)^(1/2),x). The user applies the substitution 3x+1=3secT and derives the integral to yield (1/3)|(3x+1)/3+(9x^2+6x-8)^(1/2)/3|. However, the textbook presents a different solution: (1/3)|(3x+1)+(9x^2+6x-8)^(1/2)|. The discrepancy lies in the simplification of the terms after substitution.
PREREQUISITESStudents studying calculus, particularly those focusing on integration techniques, and educators looking for examples of trigonometric substitution applications.
nameVoid said:Homework Statement
I(1/(9x^2+6x-8)^(1/2),x)
I( 1/( (3x+1)^2-9 )^(1/2),x )
3x+1=3secT
dx=secTtanTdT
I( (secTtanT)/3tanT,T )
(1/3)I( secT,T)
(1/3)|secT+tanT|+C
secT=(3x+1)/3
tanT=(9x^2+6x-8)^(1/2)/3
solution: (1/3)|(3x+1)/3+(9x^2+6x-8)^(1/2)/3|
my book is showing :(1/3)|(3x+1)+(9x^2+6x-8)^(1/2)|