∫(1 + 3x)x^2 dx
Understand the Problem
The question is asking us to find the integral of the function (1 + 3x)x^2 with respect to x. This involves applying integration techniques to solve the integral.
Answer
$$ \frac{x^3}{3} + \frac{3x^4}{4} + C $$
Answer for screen readers
The final answer is: $$ \frac{x^3}{3} + \frac{3x^4}{4} + C $$
Steps to Solve
- Expand the Integrand
Begin by expanding the integrand ( (1 + 3x)x^2 ): [ (1 + 3x)x^2 = x^2 + 3x^3 ]
- Set Up the Integral
Now, express the integral with the expanded form: [ \int (1 + 3x)x^2 , dx = \int (x^2 + 3x^3) , dx ]
- Integrate Each Term
Integrate each term separately: [ \int x^2 , dx = \frac{x^3}{3} ] [ \int 3x^3 , dx = 3 \cdot \frac{x^4}{4} = \frac{3x^4}{4} ]
- Combine the Results
Combine the results from the integration: [ \int (1 + 3x)x^2 , dx = \frac{x^3}{3} + \frac{3x^4}{4} + C ]
- Final Expression
The final expression for the integral is: [ \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 involves applying basic integration rules and polynomial expansion. Remember that ( C ) is the constant of integration that represents an indefinite integral.
Tips
- Forgetting to Expand: One common mistake is neglecting to expand the polynomial before integrating. Always simplify first.
- Missing the Constant of Integration: It's crucial not to forget the constant ( C ) when calculating indefinite integrals.
