∫(1 + 3x)x² dx

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Understand the Problem

The question is asking us to evaluate the integral of the expression (1 + 3x)x² with respect to x. This involves using techniques from calculus to find the antiderivative of the given function.

Answer

The integral evaluated is \(\frac{x^3}{3} + \frac{3x^4}{4} + C\).
Answer for screen readers

The final result of the integral is:

[ \frac{x^3}{3} + \frac{3x^4}{4} + C ]

Steps to Solve

  1. Expand the integrand

First, expand the expression ((1 + 3x)x^2) to make integration easier.

[ (1 + 3x)x^2 = x^2 + 3x^3 ]

  1. Set up the integral

Now, substitute the expanded expression into the integral:

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

  1. Integrate each term

Now, integrate each term separately:

  • The integral of (x^2) is (\frac{x^3}{3}).

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

  • The integral of (3x^3) is (\frac{3x^4}{4}).

[ \int 3x^3 , dx = \frac{3x^4}{4} ]

  1. Combine the results

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 result of the integral is:

[ \frac{x^3}{3} + \frac{3x^4}{4} + C ]

More Information

This integral represents the area under the curve defined by the function ((1 + 3x)x^2) with respect to (x). The constant (C) accounts for all possible vertical shifts of the antiderivative.

Tips

  • Forgetting to include the constant of integration (C) when expressing the final answer. Always remember that the antiderivative is not unique.
  • Miscalculating the powers of (x) when integrating. Double-check to ensure the power rule is applied correctly.

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