What is the integral of x squared over x divided by 6?
Understand the Problem
The question is asking for the integral of the function x squared divided by 6. To solve this, we will apply the basic rules of integration for polynomial functions.
Answer
The integral of \( \frac{x^2}{6} \) is \( \frac{x^3}{18} + C \).
Answer for screen readers
The integral of ( \frac{x^2}{6} ) is: $$ \frac{x^3}{18} + C $$
Steps to Solve
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Set Up the Integral We need to integrate the function ( \frac{x^2}{6} ). This can be expressed as: $$ \int \frac{x^2}{6} , dx $$
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Factor Out the Constant Since ( \frac{1}{6} ) is a constant, we can factor it out of the integral: $$ \frac{1}{6} \int x^2 , dx $$
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Integrate the Polynomial Now we integrate ( x^2 ). The rule for integrating ( x^n ) is: $$ \int x^n , dx = \frac{x^{n+1}}{n+1} + C $$ For ( n=2 ), this becomes: $$ \int x^2 , dx = \frac{x^{2+1}}{2+1} + C = \frac{x^3}{3} + C $$
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Combine the Results Substituting back into our equation gives: $$ \frac{1}{6} \left( \frac{x^3}{3} + C \right) $$ This simplifies to: $$ \frac{x^3}{18} + \frac{C}{6} $$
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Final Form The resulting expression for the integral is: $$ \frac{x^3}{18} + C' $$ where ( C' ) is a new constant of integration which can be represented as ( \frac{C}{6} ).
The integral of ( \frac{x^2}{6} ) is: $$ \frac{x^3}{18} + C $$
More Information
The process of integrating a polynomial function is a fundamental concept in calculus. It involves increasing the exponent of the polynomial by 1 and dividing by the new exponent. This result is used in physics, engineering, and other fields to determine areas under curves and total quantities.
Tips
- Forgetting to factor out the constant when integrating.
- Not applying the correct power rule for integration.
- Confusing the constant of integration ( C ) with specific numerical values.
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