∫(1 + 3x)x² dx

Understand the Problem

The question is asking to find the integral of the expression (1 + 3x)x² with respect to x. This involves applying techniques of integration to solve the problem.

Answer

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

Steps to Solve

Expand the expression

First, we need to expand the integrand $(1 + 3x)x^2$:

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

Set up the integral

Now we can rewrite our integral with the expanded expression:

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

Integrate each term

We'll integrate each term separately:

For $x^2$, the integral is:

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

For $3x^3$, the integral is:

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

Combine the results

Now, combine the results from the integrations:

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

where $C$ is the constant of integration.

The final answer is:

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

More Information

This result can be contextualized in the study of calculus, particularly in the application of the power rule of integration. The power rule is fundamental for integrating polynomials, allowing quick calculation of areas under curves defined by polynomial functions.

Tips

Neglecting to distribute the terms correctly when expanding the expression.
Forgetting to add the constant of integration $C$ at the end of the solution.
Incorrectly applying the power rule by failing to adjust the exponent properly.

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