∫(1 + 3x)x² dx
Understand the Problem
The question is asking to evaluate the integral of the function (1 + 3x)x² with respect to x. We will need to apply integration techniques to solve it.
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
First, we need to expand the function ( (1 + 3x)x^2 ):
$$(1 + 3x)x^2 = x^2 + 3x^3$$
- Set up the integral
Now we can rewrite the integral using the expanded form:
$$ \int (1 + 3x)x^2 , dx = \int (x^2 + 3x^3) , dx $$
- Integrate each term
Next, we can 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
Now, we combine the results of the integrals:
$$ \int (x^2 + 3x^3) , dx = \frac{x^3}{3} + \frac{3x^4}{4} + C $$
where ( C ) is the constant of integration.
The final answer is:
$$ \frac{x^3}{3} + \frac{3x^4}{4} + C $$
More Information
This integral represents the area under the curve of the function ( (1 + 3x)x^2 ) with respect to ( x ). Integrating polynomial functions is straightforward using the power rule.
Tips
- Forgetting to expand the expression before integrating can lead to errors.
- Misapplying the power rule, particularly in determining the correct coefficients during integration.
- Omitting the constant of integration ( C ) at the end.
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