please solve and show all steps. 11) $int(x + 7)cdotcos(x)mathrm{d}x$

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 $u = x + 7$ and $\mathrm{d}v=\cos(x)\mathrm{d}x$. Then $\mathrm{d}u=\mathrm{d}x$ and $v=\sin(x)$.

Step2: Substitute into the formula

$\int(x + 7)\cos(x)\mathrm{d}x=(x + 7)\sin(x)-\int\sin(x)\mathrm{d}x$

Step3: Integrate $\int\sin(x)\mathrm{d}x$

We know that $\int\sin(x)\mathrm{d}x=-\cos(x)+C$.

Step4: Simplify the result

$(x + 7)\sin(x)-\int\sin(x)\mathrm{d}x=(x + 7)\sin(x)-(-\cos(x))+C=(x + 7)\sin(x)+\cos(x)+C$

Answer:

$(x + 7)\sin(x)+\cos(x)+C$