evaluate the integral. (use c for the constant of integration.)\n\\(\\int7x\\cos(6x)dx\\)

Explanation:

Step1: Apply integration - by - parts formula

The integration - by - parts formula is $\int u\mathrm{d}v=uv-\int v\mathrm{d}u$. Let $u = 7x$ and $\mathrm{d}v=\cos(6x)\mathrm{d}x$. Then $\mathrm{d}u = 7\mathrm{d}x$ and $v=\frac{1}{6}\sin(6x)$.

Step2: Substitute into the formula

$\int 7x\cos(6x)\mathrm{d}x=7x\cdot\frac{1}{6}\sin(6x)-\int\frac{1}{6}\sin(6x)\cdot7\mathrm{d}x$.

Step3: Evaluate the remaining integral

$\int\frac{7}{6}\sin(6x)\mathrm{d}x=-\frac{7}{36}\cos(6x)+C$. So, $\int 7x\cos(6x)\mathrm{d}x=\frac{7}{6}x\sin(6x)+\frac{7}{36}\cos(6x)+C$.

Answer:

$\frac{7}{6}x\sin(6x)+\frac{7}{36}\cos(6x)+C$