Dada a integral: [tex]\int x \cdot \cos x \, dx[/tex]
Consideremos
[tex]f(x) = x[/tex] e [tex]g'(x) = \cos x[/tex],
onde
[tex]\int f(x) \cdot g'(x) \, dx = f(x) \cdot g(x) - \int f'(x) \cdot g(x) \, dx[/tex]
Segue,
[tex]\begin{cases} f(x) = x \Rightarrow f'(x) = 1 \\ g'(x) = \cos x \Rightarrow g(x) = \sin x \end{cases} \\\\ \int f(x) \cdot g'(x) \, dx = f(x) \cdot g(x) - \int f'(x) \cdot g(x) \, dx \\\\ \int x \cdot \cos x \, dx = x \cdot \sin x - \int 1 \cdot \sin x \, dx \\\\ \int x \cdot \cos x \, dx = x \cdot \sin x - \int \sin x \, dx \\\\ \int x \cdot \cos x \, dx = x \cdot \sin x - \left [ - \cos x \right ] \\\\ \boxed{\int x \cdot \cos x \, dx = \boxed{\boxed{x \cdot \sin x + \cos x + C}}}[/tex]
