Mathematics

Question

Integrate e^x(sin(x) cos(x))

1 Answer

  • [tex]I=\int e^x(\sin(x)\cos(x))dx=\int e^x(\frac{1}{2}\sin(2x))dx=\frac{1}{2}\int e^x\sin(2x)dx[/tex]

    [tex]\text{If }u=\sin(2x)\to du=2\cos(2x)dx~\text{and}~dv=e^xdx\to v=e^x:\\\\ \text{Using }\int u\,dv=uv-\int v\,du:\\\\ I=\frac{1}{2}(e^x\sin(2x)-\int e^x(2\cos(2x))dx)\\\\ 2I=e^x\sin(2x)-2\underbrace{\int e^x\cos(2x)dx}_{I_2}[/tex]

    Looking for [tex]I_2[/tex]:

    [tex]\text{If}~u=\cos(2x)\to du=-2\sin(2x)dx~\text{and}~dv=e^xdx\to v=e^x:\\\\ I_2=e^x\cos(2x)-\int e^x(-2\sin(2x))dx\\\\ I_2=e^x\cos(2x)+2\int e^x(\sin(2x))dx\\\\ I_2=e^x\cos(2x)+2\int e^x(2\sin(x)\cos(x))dx\\\\ I_2=e^x\cos(2x)+4\int e^x(\sin(x)\cos(x))dx=e^x\cos(2x)+4I[/tex]

    Replacing:

    [tex]2I=e^x\sin(2x)-2I_2\iff\\\\2I=e^x\sin(2x)-2(e^x\cos(2x)+4I)\iff\\\\ 2I=e^x\sin(2x)-2e^x\cos(2x)-8I\iff\\\\ 10I=e^x\sin(2x)-2e^x\cos(2x)\iff\\\\ I=\dfrac{e^x}{10}(\sin(2x)-2\cos(2x))\\\\ \boxed{\int e^x(\sin(x)\cos(x))dx=\dfrac{e^x}{10}(\sin(2x)-2\cos(2x))+C}[/tex]
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