∫(1 + 3x)x² dx

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Understand the Problem

The question is asking to evaluate the integral of the function (1 + 3x)x² with respect to x. We will need to apply integration techniques to solve it.

Answer

$$ \frac{x^3}{3} + \frac{3x^4}{4} + C $$
Answer for screen readers

The final answer is:

$$ \frac{x^3}{3} + \frac{3x^4}{4} + C $$

Steps to Solve

  1. Expand the integrand

First, we need to expand the function ( (1 + 3x)x^2 ):

$$(1 + 3x)x^2 = x^2 + 3x^3$$

  1. Set up the integral

Now we can rewrite the integral using the expanded form:

$$ \int (1 + 3x)x^2 , dx = \int (x^2 + 3x^3) , dx $$

  1. Integrate each term

Next, we can integrate each term separately:

$$ \int x^2 , dx = \frac{x^3}{3} $$

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

  1. Combine the results

Now, we combine the results of the integrals:

$$ \int (x^2 + 3x^3) , dx = \frac{x^3}{3} + \frac{3x^4}{4} + C $$

where ( C ) is the constant of integration.

The final answer is:

$$ \frac{x^3}{3} + \frac{3x^4}{4} + C $$

More Information

This integral represents the area under the curve of the function ( (1 + 3x)x^2 ) with respect to ( x ). Integrating polynomial functions is straightforward using the power rule.

Tips

  • Forgetting to expand the expression before integrating can lead to errors.
  • Misapplying the power rule, particularly in determining the correct coefficients during integration.
  • Omitting the constant of integration ( C ) at the end.

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