Question

Exercitiul 13(a d si f). Multumesc anticipat

Answer 1

a)   faci    substitutia    1+e^x=y=>e^xdx=dy

integrala     devine

∫dy/ y=lny+C

revii    la      substitutie

F(x)=ln(1+e^x)+c

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d)F(x)=∫lnxdx/x

faci    substitutia lnx=y   dx/x=dy   integrala   devine

F(y)=∫dy=y+C

Revii    la     substitutie

F(x)=lnx+c

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I=∫cosxdx/(1+sin²x)       x∈[0,π/2]

Faci     substitutia sinx=t =.>.cosxdx=dt

Schimbi   limitele    de    integrare

x=0 sin0=t=0

x=π/2 sinπ/2=t=1

I=∫dt/(1+t²)    t∈[0,1]

I=arctgt /o↑1=arctg1-arctg 0=π/4-0=π/4