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

  1. 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

$$ 6x $$

  1. Set Up the Integral

Now, set up the integral of the simplified expression. You want to find

$$ \int 6x , dx $$

  1. 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, $n=1$:

$$ \int 6x , dx = 6 \cdot \frac{x^{1+1}}{1+1} + C = 6 \cdot \frac{x^2}{2} + C $$

  1. 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.
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