∫(1 + 3x)x² dx
Understand the Problem
The question is asking for the integral of the function (1 + 3x)x² with respect to x. This involves applying the rules of integration, potentially involving polynomial expansion.
Answer
The integral is $$ \frac{x^3}{3} + \frac{3x^4}{4} + C $$
Answer for screen readers
The integral is
$$ \frac{x^3}{3} + \frac{3x^4}{4} + C $$
Steps to Solve
- Expand the integrand
To simplify the integration process, first expand the expression ( (1 + 3x)x^2 ):
[ (1)(x^2) + (3x)(x^2) = x^2 + 3x^3 ]
- Set up the integral
Now, write the integral with the expanded expression:
[ \int (x^2 + 3x^3) , dx ]
- Integrate each term
Integrate the polynomial term by term. The integral of ( x^n ) is given by ( \frac{x^{n+1}}{n+1} ):
[ \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 of the integration:
[ \int (x^2 + 3x^3) , dx = \frac{x^3}{3} + \frac{3x^4}{4} + C ]
where ( C ) is the constant of integration.
The integral is
$$ \frac{x^3}{3} + \frac{3x^4}{4} + C $$
More Information
This integral combines different polynomial terms. The technique of expansion before integration simplifies the application of the integral rules. The constant ( C ) represents any constant that could be added to the function.
Tips
- Forgetting to distribute correctly during the expansion.
- Not adding the constant of integration ( C ) at the end.
- Incorrectly applying the power rule of integration.
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