How do you evaluate the integral #int sqrt(4+x^2)#?
1 Answer
Explanation:
You can use identity:
#cosh^2 y - sinh^2 y = 1 implies cosh^2y = 1 + sinh^2 y#
So let:
-
#x^2 = 4 sinh^2 y# -
#implies 2 x \ dx = 8 sinh y \ cosh y \ dy#
Considering:
-
#color(red)(sinh 2y = 2 sinh y cosh y )# -
#x^2 = 4 sinh^2 y implies color(red)( x = 2 sinh y) color(red)(implies y = sinh^(-1) (x/2)) # -
# cosh^2y = 1 + sinh^2 y implies color(red)(cosh y = sqrt(1 + x^2/4) )#
Then