Question

quem souber ajude. ...........

Answer 1

Resolvendo a integral por partes ($\int\limits {u} \, dv = uv- \int\limits {v} \, du$)

$\int\limits {te^{3t}} \, dt$


$u= t$                     $dv= e^{3t}$

$du= dt$                  $v= \int\limits {dv} \, dt$


$v= \int\limits e^{3t} \, dt u= 3t du= 3dt dt = du/3 v= \int\limits {e^u} \, \frac{du}{3}$



$v= \frac{1}{3}e^{3t}$

$uv- \int\limits {v} \, du$

$t. \frac{1}{3}e^{3t}- \int\limits { \frac{1}{3}e^{3t} } \, dx$

$\frac{1}{3}e^{3t}t- \frac{1}{3} \int\limits {e^{3t} } \, dx$

$\frac{1}{3}e^{3t}t- \frac{1}{3}. \frac{1}{3} {e^{3t} } + C$

$\frac{1}{3}e^{3t}t- \frac{1}{9} {e^{3t} } + C$


Colocando o fator comum em evidência:

$\frac{1}{3}e^{3t}t- \frac{1}{9} {e^{3t} } = \boxed{\boxed{e^{3t}( \frac{t}{3}- \frac{1}{9} )+C}}$

Answer 2

Olá!

Coluna I: termo a derivar

Coluna II: termo a integrar


Coluna I _____________ Coluna II

t ___________________ e^{3t}

1 ___________________ 2^{3t} . 1/3

0 ___________________ 2^{3t} . 1/9