∫(1 + 3x)x^2 dx

Question image

Understand the Problem

The question is asking us to find the integral of the function (1 + 3x)x^2 with respect to x. This involves applying integration techniques to solve the integral.

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

Begin by expanding the integrand ( (1 + 3x)x^2 ): [ (1 + 3x)x^2 = x^2 + 3x^3 ]

  1. Set Up the Integral

Now, express the integral with the expanded form: [ \int (1 + 3x)x^2 , dx = \int (x^2 + 3x^3) , dx ]

  1. Integrate Each Term

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

Combine the results from the integration: [ \int (1 + 3x)x^2 , dx = \frac{x^3}{3} + \frac{3x^4}{4} + C ]

  1. Final Expression

The final expression for the integral is: [ \frac{x^3}{3} + \frac{3x^4}{4} + C ]

The final answer is: $$ \frac{x^3}{3} + \frac{3x^4}{4} + C $$

More Information

This integral involves applying basic integration rules and polynomial expansion. Remember that ( C ) is the constant of integration that represents an indefinite integral.

Tips

  • Forgetting to Expand: One common mistake is neglecting to expand the polynomial before integrating. Always simplify first.
  • Missing the Constant of Integration: It's crucial not to forget the constant ( C ) when calculating indefinite integrals.
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