∫(1 + 3x)x² dx

Question image

Understand the Problem

The question is asking for the integration of the expression (1 + 3x)x² with respect to x. This involves finding the antiderivative of the function, which may require expanding the expression and then applying integration rules.

Answer

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

$$\int (1 + 3x)x^2 , dx = \frac{x^{3}}{3} + \frac{3x^{4}}{4} + C$$

Steps to Solve

  1. Expand the Expression

First, distribute $x^2$ in the integrand:

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

  1. Set Up the Integral

Now, rewrite the integral with the expanded expression:

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

  1. Integrate Each Term

Next, integrate each term separately:

  • The integral of $x^n$ is given by the formula:

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

  • For $x^2$, the integral is:

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

  • For $3x^3$, the integral is:

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

  1. Combine the Results

Combine the results of the integrals:

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

  1. Final Expression

The final expression for the integral is:

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

$$\int (1 + 3x)x^2 , dx = \frac{x^{3}}{3} + \frac{3x^{4}}{4} + C$$

More Information

The resulting expression is the antiderivative of the given function. When integrating polynomial functions, expanding the product before integrating can simplify the process significantly. The constant ( C ) represents the constant of integration.

Tips

  • Forgetting to include the constant of integration ( C ) at the end of the problem.
  • Making errors during the distribution step or when applying the power rule for integration.

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