Evaluate the integral of (x + 3) with respect to x.

Step-by-step solution

1. 1

   Split the integral using the sum rule. Integration is linear, so a sum can be handled one term at a time:

   $\int(x+3)\,dx=\int x\,dx+\int 3\,dx$

2. 2

   Integrate the linear term with the power rule. Here $n=1$, so $n+1=2$:

   $\int x\,dx=\frac{x^{2}}{2}$

3. 3

   Integrate the constant. A constant $k$ is $kx^0$, and the power rule sends it to $kx$:

   $\int 3\,dx=3x$

   The intuition: the function whose slope is constantly $3$ is a line of gradient $3$.

4. 4

   Combine, adding a single constant of integration. Two separate constants would collapse into one anyway:

   $\int(x+3)\,dx=\frac{x^2}{2}+3x+C$

5. 5

   Check by differentiating. $\tfrac{d}{dx}\left(\tfrac{x^2}{2}+3x+C\right)=x+3$ $\checkmark$. A numerical check at $x=1.3$: the difference quotient of $\tfrac{x^2}{2}+3x$ gives $4.300$, and the integrand is $1.3+3=4.3$ $\checkmark$.