∫(1 + 3x)x² dx

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

Steps to Solve

Expand the expression

First, we expand the expression inside the integral:

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

Set up the integral

Now, we write the integral with the expanded expression:

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

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 $3x^3$:

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

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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