What is the integral of x squared over (x divided by 6)?

Understand the Problem

The question is asking for the integral of (x squared) / (x/6). To solve this, we first simplify the expression and then apply the rules of integration.

Answer

The integral of $ \frac{x^2}{\frac{x}{6}} $ is $ 3x^2 + C $.

Steps to Solve

Simplify the Expression

First, simplify the integrand. We have:

$$ \frac{x^2}{\frac{x}{6}} $$

This can be simplified by multiplying the numerator by the reciprocal of the denominator:

$$ = x^2 \times \frac{6}{x} $$

This gives us:

$$ = 6x $$

Set Up the Integral

Now, we can set up the integral with the simplified expression:

$$ \int 6x , dx $$

Integrate the Expression

Now, apply the power rule of integration. The power rule states that:

$$ \int x^n , dx = \frac{x^{n+1}}{n+1} + C $$

For our integral:

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

Simplify the Result

Now, simplify the expression:

$$ = 3x^2 + C $$

This is the final result of the integration.

The integral of ( \frac{x^2}{\frac{x}{6}} ) is ( 3x^2 + C ).

More Information

This integration uses basic algebra and the power rule for integration. The constant ( C ) represents the constant of integration which is added because integral equations can have infinite solutions due to varying constants.

Tips

Forgetting to simplify the expression before integrating, which can lead to more complex calculations.
Not applying the integral constant ( C ) correctly, which is crucial in indefinite integrals.

AI-generated content may contain errors. Please verify critical information

Thank you for voting!