Question

4) (2,0) Calcule a integral pelo método de integração por partes:

a) (1,0) ∫ ln(x) dx
b) (1,0) ∫ (3x).sen(x)

Answer 1

Resposta:

Passo - a - passo

$a)\displaystyle\int lnxdx=\\\\lnx=u\implies\frac{1}{x}dx=du~~e~~dx=dv~\implies \displaystyle\int dx=\displaystyle\int dv \implies x=v\\\\\displaystyle\int udv=uv-\displaystyle\int vdu\\\\\displaystyle\int lnxdx=lnx*x-\displaystyle\int x*\frac{1}{x}dx\\\\ \displaystyle\int lnxdx=xlnx-\displaystyle\int dx\\\\\displaystyle\int lnxdx=xlnx-x+C$

$b)\displaystyle\int 3x~senx~dx\\\\3x=u\implies3dx=du \imploes ~e~senxdx=dv\implies\displaystyle\int senxdx=\displaystyle\int dv \implies-cosx=v\\\\\displaystyle\int udv=uv-\displaystyle\int vdu\\\\\displaystyle\int 3x~senx~dx=3x*(-cosx)-\displaystyle\int -cosx.3dx\\\\\displaystyle\int 3x~senx~dx=-3xcosx+3senx+C$