∫(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. This involves applying integration techniques to find the antiderivative of the given expression.
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
- Distribute the expression
First, we need to expand the integrand ( (1 + 3x)x^2 ).
[ (1 + 3x)x^2 = x^2 + 3x^3 ]
- Write the integral in expanded form
Now, we rewrite the integral:
[ \int (1 + 3x)x^2 , dx = \int (x^2 + 3x^3) , dx ]
- Apply the power rule of integration
Using the power rule, we integrate each term separately. The power rule states that:
[ \int x^n , dx = \frac{x^{n+1}}{n+1} + C ]
For the first term ( x^2 ):
[ \int x^2 , dx = \frac{x^{2+1}}{2+1} = \frac{x^3}{3} ]
For the second term ( 3x^3 ):
[ \int 3x^3 , dx = 3 \cdot \frac{x^{3+1}}{3+1} = \frac{3x^4}{4} ]
- Combine the results
Now, we combine the results of our integrations:
[ \int (x^2 + 3x^3) , dx = \frac{x^3}{3} + \frac{3x^4}{4} + C ]
- Final expression
To write our final answer in a standard form, we can express it as:
[ \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 shows the application of the power rule in calculus, and integrates a polynomial function. The presence of constants in the integration process emphasizes that there can be multiple antiderivatives differing by a constant.
Tips
- Forgetting to add the constant of integration ( C ) after finding the antiderivative.
- Misapplying the power rule (for example, incorrectly calculating ( n+1 ) for the exponent).
