∫(1 + 3x)x² dx
Understand the Problem
The question is asking us to evaluate the integral of the function (1 + 3x)x². This involves using integration techniques, such as polynomial expansion and then applying the power rule of integration.
Answer
The integral evaluates to $$ \frac{x^{3}}{3} + \frac{3x^{4}}{4} + C $$
Answer for screen readers
The result of the integral is
$$ \frac{x^{3}}{3} + \frac{3x^{4}}{4} + C $$
Steps to Solve
- Expand the integrand
First, distribute ( x^2 ) across the terms in the parentheses:
$$ (1 + 3x)x^2 = x^2 + 3x^3 $$
- Set up the integral
Now we can rewrite the integral using the expanded form:
$$ \int (x^2 + 3x^3) , dx $$
- Apply the power rule of integration
Using the power rule, which states that ( \int x^n , dx = \frac{x^{n+1}}{n+1} + C ), integrate each term:
For ( x^2 ):
$$ \int x^2 , dx = \frac{x^{3}}{3} $$
For ( 3x^3 ):
$$ \int 3x^3 , dx = 3 \cdot \frac{x^{4}}{4} = \frac{3x^{4}}{4} $$
- Combine the results
Now combine the results of the integrals:
$$ \int (x^2 + 3x^3) , dx = \frac{x^{3}}{3} + \frac{3x^{4}}{4} + C $$
- Simplify if necessary
Sometimes, it might be helpful to express the final answer in a single fraction, but here we leave it as is for clarity.
The result of the integral is
$$ \frac{x^{3}}{3} + \frac{3x^{4}}{4} + C $$
More Information
The process of expanding the integrand and then applying the power rule is a fundamental technique in calculus. It allows you to solve many integrals involving polynomials easily. Remember that ( C ) represents the constant of integration, which is essential in indefinite integrals.
Tips
- Forgetting to include the constant of integration ( C ).
- Misapplying the power rule, such as failing to increase the exponent correctly.
- Ignoring to distribute correctly during the expansion step.
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