∫(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 $$\frac{x^3}{3} + \frac{3x^4}{4} + C$$.
Answer for screen readers
The final answer 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.
