całka Cotoznaczy: Jak policzyć całkę: ∫√1+²_x ?

Mariusz: √1+²_x=t

	2
1+		=t2
	x

2
	=t2−1
x

1		t2−1
	=
x		2

	2
x=
	t2−1

	0*(t2−1)−2*2t
dx=		dt
	(t2−1)2

	4t
dx=−		dt
	(t2−1)2

	4t2
∫−		dt
	(t2−1)2

(t+1)+(t−1)=2t (t+1)²+2(t+1)(t−1)+(t−1)²=4t² (t+1)²+(t−1)²+[(t+1)(t−1)][(t+1)−(t−1)]=4t² (t+1)²+(t−1)²+(t+1)²(t−1)−(t+1)(t−1)²=4t²

	(t+1)2+(t−1)2+(t+1)2(t−1)−(t+1)(t−1)2
−∫		dt
	(t−1)2(t+1)2

	1		1		1		1
−(∫		dt+∫		dt+∫		dt−∫		dt)
	(t−1)2		(t+1)2		t−1		t+1

	1		1		t+1
=		+		+ln|		|+C
	t−1		t+1		t−1

	t+1+t−1		t2−1+2t+2
=		+ln|		|+C
	t2−1		t2−1

	2t		t2−1+2t+2
=		+ln|		|+C
	t2−1		t2−1

=x√1+²_x+ln|1+x+x√1+²_x|+C

jc:

	2
t2=1+
	x

	2
x=
	t2−1

2

2t

dt

2t

1

t−1

całka= ∫t(

)' dt =

− ∫

=

−

ln|

|

t2−1

t2−1

t2−1

t2−1

2

t+1

Po co różniczkować, skoro zaraz potem całkujemy?