Integralen

[tiger31]

hey,

Weet iemand hoe je aan deze oplossingen komt voor volgende integralen?

integraal (1+sqrt(1-x^2)-x^2)/(1-x^2)dx = x+asin(x)+C

integraal (4x+3sqrt(x^2-1))/(xsqrt(x^2-1)) dx =
4 ln|x+sqrt(x^2-1)|+3ln|x|+C

grts

[TD]

tiger31 schreef op 08-03-2006 @ 21:56 :
integraal (1+sqrt(1-x^2)-x^2)/(1-x^2)dx = x+asin(x)+C

integraal (4x+3sqrt(x^2-1))/(xsqrt(x^2-1)) dx =
4 ln|x+sqrt(x^2-1)|+3ln|x|+C

grts

1)
Int (1+sqrt(1-x²)-x²)/(1-x²) dx
Int (1-x²)/(1-x²)+sqrt(1-x²))/(1-x²) dx
Int 1+1/sqrt(1-x²) dx
x + arcsin(x) + C


2)
Int (4x+3sqrt(x²-1))/(xsqrt(x²-1)) dx
Int 4/(sqrt(x²-1)) dx + Int 3/x dx
Int 4/(sqrt(x²-1)) dx + 3ln|x|

Voor de eerste integraal: y = x+sqrt(x²-1) <=> dy = x+sqrt(x²-1)/sqrt(x²-1) dx

=> Int 4/(sqrt(x²-1)) dx = 4 Int dy/y = 4ln|y| + C -> 4ln|x+sqrt(1-x²)|

Dus samen: 4ln|x+sqrt(1-x²)| + 3ln|x| + C.

[tiger31]

Bedankt!

[TD]

Graag gedaan