Question: integrate $\int Ce^x\left(x+1\right)\,dx$ please with steps
1. **State the problem:** We need to find the integral $$\int Ce^x(x+1)\,dx$$ where $C$ is a constant.
2. **Recall the formula and rules:** The integral of a product involving $e^x$ and a polynomial can be solved using integration by parts or recognizing patterns.
3. **Rewrite the integral:** $$\int Ce^x(x+1)\,dx = C \int e^x(x+1)\,dx$$ since $C$ is a constant, it can be factored out.
4. **Use integration by parts:** Let
- $u = x+1$ so that $du = dx$
- $dv = e^x dx$ so that $v = e^x$
5. **Apply integration by parts formula:** $$\int u\,dv = uv - \int v\,du$$
6. **Calculate:**
$$\int e^x(x+1) dx = (x+1)e^x - \int e^x dx$$
7. **Integrate remaining integral:**
$$\int e^x dx = e^x + C_1$$ (where $C_1$ is the integration constant)
8. **Substitute back:**
$$\int e^x(x+1) dx = (x+1)e^x - e^x + C_1 = xe^x + C_1$$
9. **Multiply by $C$:**
$$\int Ce^x(x+1) dx = Cxe^x + CC_1 = Cxe^x + C_2$$ where $C_2$ is the general constant of integration.
**Final answer:**
$$\boxed{\int Ce^x(x+1) dx = Cxe^x + C_2}$$