Bonjour Design971
[tex]\int t.e^{3t}\,dt=?\\\\\boxed{u(t)=t\Longrightarrow u'(t)=1}\\\\\boxed{v'(t)=e^{3t}\Longrightarrow v(t)=\dfrac{1}{3}e^{3t}}\\\\\\\int t.e^{3t}\,dt=t\times\dfrac{1}{3}e^{3t}-\int 1\times\dfrac{1}{3}e^{3t}\,dt\\\\\int t.e^{3t}\,dt=\dfrac{1}{3}t.e^{3t}-\dfrac{1}{3}\int e^{3t}\,dt\\\\\int t.e^{3t}\,dt=\dfrac{1}{3}t.e^{3t}-\dfrac{1}{3}\times\dfrac{1}{3}e^{3t}+C\\\\\boxed{\int t.e^{3t}\,dt=\dfrac{1}{3}t.e^{3t}-\dfrac{1}{9}e^{3t}+C}\\\\\boxed{\int t.e^{3t}\,dt=\dfrac{1}{9}e^{3t}(3t-1)+C}[/tex]
