Evaluate the integral of (x^3 + 6) with respect to x.

Step-by-step solution

1. 1

   Use linearity to separate the two terms.

   $\int\left(x^3+6\right)dx=\int x^3\,dx+\int 6\,dx$

2. 2

   State the power rule and apply it with $n=3$.

   $\int x^n\,dx=\frac{x^{n+1}}{n+1}+C\quad(n\neq-1)$

   $\int x^3\,dx=\frac{x^{4}}{4}$

   The exponent goes up by one — a frequent slip is to lower it, which is the differentiation pattern instead.

3. 3

   Integrate the constant term.

   $\int 6\,dx=6x$

4. 4

   Write the general antiderivative. One constant covers the whole family:

   $\int\left(x^3+6\right)dx=\frac{x^4}{4}+6x+C$

5. 5

   Verify by differentiating. $\tfrac{d}{dx}\left(\tfrac{x^4}{4}+6x\right)=x^3+6$ $\checkmark$. Numerically at $x=1.1$ the difference quotient gives $7.331$, and $1.1^3+6=7.331$ $\checkmark$.