Question

Evaluate \[ \int_{-1}^{2} \left( x^3 - |x| \right) dx. \]

Hint:

When dealing with absolute value functions in integrals, break the integral into pieces where the expression inside the absolute value is either positive or negative.

Answer 1

Step 1: Break the integral into regions based on the absolute value.
We need to split the integral at \( x = 0 \) because the absolute value function \( |x| \) behaves differently for positive and negative values of \( x \). Thus, we can write: \[ \int_{-1}^{2} \left( x^3 - |x| \right) dx = \int_{-1}^{0} \left( x^3 + x \right) dx + \int_{0}^{2} \left( x^3 - x \right) dx. \]

Step 2: Calculate each part of the integral.
- First integral: \[ \int_{-1}^{0} \left( x^3 + x \right) dx = \left[ \frac{x^4}{4} + \frac{x^2}{2} \right]_{-1}^{0} = \left( 0 + 0 \right) - \left( \frac{(-1)^4}{4} + \frac{(-1)^2}{2} \right) = 0 - \left( \frac{1}{4} + \frac{1}{2} \right) = -\frac{3}{4}. \] - Second integral: \[ \int_{0}^{2} \left( x^3 - x \right) dx = \left[ \frac{x^4}{4} - \frac{x^2}{2} \right]_{0}^{2} = \left( \frac{16}{4} - \frac{4}{2} \right) - (0 - 0) = 4 - 2 = 2. \]

Step 3: Combine the results.
Now, add the results from both integrals: \[ \int_{-1}^{2} \left( x^3 - |x| \right) dx = -\frac{3}{4} + 2 = \frac{5}{4}. \]

Step 4: Conclusion.
Thus, the value of the integral is: \[ \boxed{\frac{5}{4}}. \]