Question

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Answer 1

4. se  aplica  formula  ∫dx/√(x²+a²)=ln(x+√x²+4)+c  x∈[0 ,1]
3.I=∫cos²dx/sin²x·cos²x-∫sin²xdx/sin²x·cos²x=∫dx/sin²x-∫dx/cos²x= (-ctgx-tgx)lπ/6↑π/3=schimbam  semnul  functiei  si  totodata  limitele  de  integrare=(ctgx+tgx)lπ/3↑π/6=
ctgπ/6+tgπ/6-ctgπ/3-tgπ/3=√3/3+√3-√3-√3/3=0
I=ln(x+√(x²+4)lo↑1  =ln(1+√5)-ln2=ln(1+√5)/2
2.  x∈[2,3] I=∫(x²-2x+1)dx/2x=∫x²dx/2x-∫2xdx/2x+∫dx/2x=∫xdx/2-∫dx+1/2∫dx/x=
(x²/4-x+1/2lnx)l2↑3=(9/4-3+1/2ln3)-(4/4-2+1/2ln2=1/4(9-12+2ln3)-(1-2+1/2ln2)=3/4+1/2ln3+1-1/2ln2=7/4+1/2ln3/2