Question

Integral indefinida de ∫ x.cos x dx

Answer 1

Resposta:   $\displaystyle\int x\cos x\,dx= x\,\mathrm{sen\,}x+\cos x+C.$

Explicação passo a passo:

Calcular a integral indefinida

    $\displaystyle\int x\cos x\,dx$

Usaremos integração por partes:

    $\begin{array}{lcl}u=x&\quad\Longrightarrow\quad&du=dx\\\\ dv=\cos x\,dx&\quad\Longleftarrow\quad&v=\mathrm{sen\,}x \end{array}$

    $\begin{array}{l}\displaystyle u\,dv=uv-\int v\,du\\\\\\\displaystyle \Longrightarrow\quad\int x\cos x\,dx= x\,\mathrm{sen\,}x-\int \mathrm{sen\,}x\,dx\\\\ \displaystyle\Longleftrightarrow\quad\int x\cos x\,dx= x\,\mathrm{sen\,}x-(-\cos x)+C\\\\ \displaystyle\Longleftrightarrow\quad\int x\cos x\,dx= x\,\mathrm{sen\,}x+\cos x+C\quad\longleftarrow\quad\mathsf{resposta.}\end{array}$

Dúvidas? Comente.

Bons estudos! :-)