Sa se stabileasca in care din intervalele urmatoare se afla valoarea integralei:
I=integrala de la 0 la 1 din sqrt(1-x^2)*arctgx dx.

a) [pi/4-ln2/2,1]
b) [1,pi/2]
c) [pi/4, 3*pi/4]
d) [0,pi/4-ln2/2]
e) [pi/4-ln2/4,1]
f) [pi/4+ln2/2,2]

0<x<1 => 0<x^2<1 |(-1) =>0>-x^2>-1 => 1>1-x^2>0 => 1>sqrt(1-x^2)>0 |arctgx
=> arctgx>sqrt(1-x^2)*arctgx>0 =>0<I<int arctgx de la 0..1
int arctgx de la 0..1 =xarctgx|0..1 - 1/2*int (2x)/(1+x^2)de la 0..1=
= arctg1-1/2*ln(x^2+1)|0..1=pi/4 - 1/2*ln2
=> 0<I<pi/4 - 1/2*ln2
raspuns : d

Rasp D)