Mistake: what is a, a constant or variable? what is S1/sqr(x^2-1) from 2~3?
differentiating ln[x+sqr(x^2-a^2)] with respect to what, a or x?
Re-post this question clearly.
.
Let y= ln( x + sqrt(x^2-a^2))
y' = [1/(x+sqrt(x^2-a^2)) ] d/dx (x + sqrt(x^2-a^2) )
d/dx (x+sqrt(x^2-a^2)) = 1 + 1/(2sqrt(x^2-a^2)) d/dx (x^2-a^2) =1 + 1/(2sqrt(x^2-a^2)) (2x) = 1+ x /sqrt(x^2-a^2)
y' = [1/(x+sqrt(x^2-a^2)) ][1+ x /sqrt(x^2-a^2)]
y' = 1/(x+sqrt(x^2-a^2)) + x / ( sqrt(x^2-a^2) ( x+sqrt(x^2-a^2)))
( 1/(x+sqrt(x^2-a^2)) + x / ( sqrt(x^2-a^2) ( x+sqrt(x^2-a^2)))
(1 /(x+sqrt(x^2-a^2)) + x / (x sqrt(x^2-a^2) +x^2-a^2)
(1 /(x+sqrt(x^2-a^2)) + x /(sqrt(x^2-a^2)(x +sqrt(x^2-a^2)))
(1/(x+sqrt(x^2-a^2)) ( 1 + x/sqrt(x^2-a^2))
(1/(x+sqrt(x^2-a^2)) ( sqrt(x^2-a^2) + x) /sqrt(x^2-a^2))
(1/(x+sqrt(x^2-a^2)) ( x+sqrt(x^2-a^2)) /sqrt(x^2-a^2))
= 1/sqrt(x^2-a^2)
y' = 1/sqrt(x^2-a^2)
Integral 1/sqrt(x^2-1) dx = ln(x+sqrt(x^2-1))
(letting a=1)
F(x) = ln(x+sqrt(x^2-1))
F(3) = ln(3+sqrt(9-1) ) = ln(3+sqrt(8))
F(2) = ln(2+sqrt(4-1) ) = ln(2+sqrt(3))
F(3)-F(2) = ln(3+sqrt(8))-ln(2+sqrt(3))
=0.44579
Did you forget a dx somewhere?
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Verified answer
Mistake: what is a, a constant or variable? what is S1/sqr(x^2-1) from 2~3?
differentiating ln[x+sqr(x^2-a^2)] with respect to what, a or x?
Re-post this question clearly.
.
Let y= ln( x + sqrt(x^2-a^2))
y' = [1/(x+sqrt(x^2-a^2)) ] d/dx (x + sqrt(x^2-a^2) )
d/dx (x+sqrt(x^2-a^2)) = 1 + 1/(2sqrt(x^2-a^2)) d/dx (x^2-a^2) =1 + 1/(2sqrt(x^2-a^2)) (2x) = 1+ x /sqrt(x^2-a^2)
y' = [1/(x+sqrt(x^2-a^2)) ][1+ x /sqrt(x^2-a^2)]
y' = 1/(x+sqrt(x^2-a^2)) + x / ( sqrt(x^2-a^2) ( x+sqrt(x^2-a^2)))
( 1/(x+sqrt(x^2-a^2)) + x / ( sqrt(x^2-a^2) ( x+sqrt(x^2-a^2)))
(1 /(x+sqrt(x^2-a^2)) + x / (x sqrt(x^2-a^2) +x^2-a^2)
(1 /(x+sqrt(x^2-a^2)) + x /(sqrt(x^2-a^2)(x +sqrt(x^2-a^2)))
(1/(x+sqrt(x^2-a^2)) ( 1 + x/sqrt(x^2-a^2))
(1/(x+sqrt(x^2-a^2)) ( sqrt(x^2-a^2) + x) /sqrt(x^2-a^2))
(1/(x+sqrt(x^2-a^2)) ( x+sqrt(x^2-a^2)) /sqrt(x^2-a^2))
= 1/sqrt(x^2-a^2)
y' = 1/sqrt(x^2-a^2)
Integral 1/sqrt(x^2-1) dx = ln(x+sqrt(x^2-1))
(letting a=1)
F(x) = ln(x+sqrt(x^2-1))
F(3) = ln(3+sqrt(9-1) ) = ln(3+sqrt(8))
F(2) = ln(2+sqrt(4-1) ) = ln(2+sqrt(3))
F(3)-F(2) = ln(3+sqrt(8))-ln(2+sqrt(3))
=0.44579
Did you forget a dx somewhere?