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find ∫ 1/sqrt[x^2-0.01].dx

الخميس، 23 فبراير، 2012 التسميات:
put x = 0.1 sec(t)

then dx = 0.1sec(t).tan(t) dt .. by substitution

∫0.1/sqrt(0.01sec²(t) - 0.01) sec(t).tan(t) dt

= ∫0.1/0.1sqrt(sec²(t) - 1) sec(t).tan(t) dt

=  ∫1/sqrt(sec²(t) - 1)  sec(t).tan(t) dt

= ∫ sec(t).tan(t)/tan(t) dt

= ∫sec(t) dt

= ln|sec(t) + tan(t)| + C

but X = 0.1sec(t) 

sec(t) = 10X

tan(t) = sqrt(sec²(t) - 1)

= sqrt(100X² - 1)

therefore ∫1/sqrt(x^2-0.01)dx 

= ln|10X + sqrt(100X² - 1) | + C‏


وهناك حل آخر عن طريق الدوال الزائدية العكسية وهو اسهل بكثير من السابق :

حيث ان :   
∫1/sqrt(x² - a²) dx  = arccosh(x/a) + C
∫1/sqrt(x² - 0.01) dx  = arccosh(10x) + C               


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