∫(1 + 3x)x² dx
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
- Distribute the terms inside the integral
We first need to distribute (x^2) to both terms inside the parentheses:
$$(1 + 3x)x^2 = x^2 + 3x^3$$
- Rewrite the integral
Now we can rewrite the integral using the distributed expression:
$$\int (x^2 + 3x^3) , dx$$
- Separate the integral
Integrate each term separately:
$$\int x^2 , dx + \int 3x^3 , dx$$
- 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:
$$\int 3x^3 , dx = 3 \cdot \frac{x^{3 + 1}}{3 + 1} = 3 \cdot \frac{x^4}{4} = \frac{3x^4}{4}$$
- 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.
