Bonjour Design971
[tex]\int\limits_a^b x.e^{3x}\,dx=?\\\\\boxed{u(x)=x\Longrightarrow u'(x)=1}\\\\\boxed{v'(x)=e^{3x}\Longrightarrow v(x)=\dfrac{1}{3}e^{3x}}\\\\\\\int\limits_a^b x.e^{3x}\,dx=[x\times\dfrac{1}{3}e^{3x}]\limits_a^b-\int\limits_a^b 1\times\dfrac{1}{3}e^{3x}\,dx\\\\=[\dfrac{1}{3}x.e^{3x}]\limits_a^b-\dfrac{1}{3}\int\limits_a^be^{3x}\,dx\\\\=[\dfrac{1}{3}x.e^{3x}]\limits_a^b-\dfrac{1}{3}[\dfrac{1}{3}e^{3x}]\limits_a^b[/tex]
[tex]\\\\=[\dfrac{1}{3}x.e^{3x}]\limits_a^b-[\dfrac{1}{9}e^{3x}]\limits_a^b\\\\=[\dfrac{1}{3}x.e^{3x}-\dfrac{1}{9}e^{3x}]\limits_a^b\\\\=\dfrac{1}{9}[e^{3x}(3x-1}]\limits_a^b\\\\=\dfrac{1}{9}[e^{3b}(3b-1)}-e^{3a}(3a-1}][/tex]
