∫(1 + 3x)x² dx

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Understand the Problem

The question is asking to solve the integral of the expression (1 + 3x)x² with respect to x.

Answer

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

The final result of the integral is

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

Steps to Solve

  1. Distribute the terms inside the integral

We first need to distribute (x^2) to both terms inside the parentheses:

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

  1. Rewrite the integral

Now we can rewrite the integral using the distributed expression:

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

  1. Separate the integral

Integrate each term separately:

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

  1. Integrate each term

Now we apply the power rule for integration, which states that

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

For the first term:

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

For the second term:

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

  1. Combine the results

Now we combine the integrals:

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

The final result of the integral is

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

More Information

The integral represents the area under the curve of the function ((1 + 3x)x^2) with respect to (x). The integration process involves using the power rule, which is a fundamental method for integrating polynomial expressions.

Tips

  • Failing to distribute correctly: Make sure to multiply every term in the parentheses by (x^2).
  • Forgetting the constant (C): Always include the constant of integration in indefinite integrals.
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