What is the integral of x squared divided by x, divided by 6?
Understand the Problem
The question is asking for the integral of the expression x squared divided by x, multiplied by 6. This involves simplifying the expression before integrating it with respect to x.
Answer
$$ 3x^2 + C $$
Answer for screen readers
The final answer is
$$ 3x^2 + C $$
Steps to Solve
- Simplify the Expression
Start by simplifying the expression before integrating. The given expression is
$$ \frac{x^2}{x} \cdot 6 $$
Since $ \frac{x^2}{x} = x $, you can rewrite the expression as
$$ 6x $$
- Set Up the Integral
Now, set up the integral of the simplified expression. You want to find
$$ \int 6x , dx $$
- Integrate the Expression
To integrate $6x$, use the power rule of integration, which states:
$$ \int x^n , dx = \frac{x^{n+1}}{n+1} + C $$
where $C$ is the constant of integration. Here, $n=1$:
$$ \int 6x , dx = 6 \cdot \frac{x^{1+1}}{1+1} + C = 6 \cdot \frac{x^2}{2} + C $$
- Simplify the Result
Now, simplify the result:
$$ 6 \cdot \frac{x^2}{2} = 3x^2 $$
So, the integral is:
$$ 3x^2 + C $$
The final answer is
$$ 3x^2 + C $$
More Information
The result of the integral represents the area under the curve of the function $6x$. The constant $C$ accounts for any vertical shift in the indefinite integral, meaning there are infinitely many antiderivatives.
Tips
- Confusing the simplification step. Make sure to divide before multiplying, as this can change the expression.
- Forgetting to include the constant of integration $C$ when finding the indefinite integral.
