Jawab:
Penjelasan dengan langkah-langkah:
Menggunakan integral parsial.
Misalkan
[tex]u=e^x[/tex]⇒[tex]du=e^x dx[/tex]
[tex]dv=sin(x)dx[/tex] ⇒[tex]v=\int\limits sin(x}) \, dx =-cos(x)[/tex]
Maka
[tex]\int\limits e^xsin(x) \, dx=uv-\int v\,du \\=e^x.(-cos(x))-\int(-cos(x)).e^x \,dx\\=-e^xcos(x)+\int e^xcos(x) \,dx. ...(1)[/tex]
Kemudian gunakan parsial lagi
[tex]u=e^x[/tex] ⇒ [tex]du=e^xdx[/tex]
[tex]dv=cos(x) dx[/tex] ⇒ [tex]v=\int cos(x) \,dx =sinx[/tex]
Maka
[tex]\int e^x cos(x) \,dx = e^x . sin(x) - \int e^x sin(x) \,dx ...(2)[/tex]
Kemudian (2) digabung ke (1) diperoleh
[tex]\int e^x sin(x)\,dx=-e^xcos(x) + e^x sin(x) - \int e^x sin(x) \,dx\\\int e^x sin(x)\,dx+\int e^x sin(x)\,dx=-e^xcos(x) + e^x sin(x)\\2.\int e^x sin(x)\,dx=-e^xcos(x) + e^x sin(x)\\\int e^x sin(x)\,dx=\frac{-e^xcos(x) + e^x sin(x)}{2}[/tex]
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