∫(1 + 3x)x² dx

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Understand the Problem

The question is asking for the integral of the expression (1 + 3x)x² with respect to x. We will likely use techniques of polynomial expansion and integration to solve it.

Answer

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

The final answer is:

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

Steps to Solve

  1. Expand the expression

First, we expand the expression inside the integral:

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

  1. Set up the integral

Now, we write the integral with the expanded expression:

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

  1. Integrate each term

Next, we integrate each term separately using the power rule for integration, which states that $\int x^n , dx = \frac{x^{n+1}}{n+1} + C$:

  • For $x^2$:

$$\int x^2 , dx = \frac{x^{2+1}}{2+1} = \frac{x^3}{3}$$

  • For $3x^3$:

$$\int 3x^3 , dx = 3 \cdot \frac{x^{3+1}}{3+1} = \frac{3x^4}{4}$$

  1. Combine results and add constant of integration

Now, combine the results from the integration:

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

The final answer is:

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

More Information

This integral represents the area under the curve of the polynomial function $(1 + 3x)x^2$. The constant $C$ denotes the integration constant, accounting for the family of antiderivatives.

Tips

  • Forgetting to expand the expression before integrating.
  • Incorrectly applying the power rule (e.g., miscalculating the exponent increase).
  • Forgetting to include the constant of integration.

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