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An antiderivative is also known as an indefinite integral. It is a function whose derivative is the original function. Finding an antiderivative is like playing the reverse game of differentiation. For example, if we have a function like \( x^3 \), its antiderivative is \( \frac{x^4}{4} \). This is because if you differentiate \( \frac{x^4}{4} \), you get back to \( x^3 \).

When dealing with polynomials like \( x^3 - 2x + 3 \), you find the antiderivative of each term separately:

• For \( x^3 \), the antiderivative is \( \frac{x^4}{4} \).
• For \( -2x \), it becomes \( -x^2 \) since the derivative of \( -x^2 \) is \( -2x \).
• For \( 3 \), it becomes \( 3x \) since the derivative of \( 3x \) is \( 3 \).

The complete antiderivative for \( x^3 - 2x + 3 \) is thus \( F(x) = \frac{x^4}{4} - x^2 + 3x \). This function will play a crucial role when using the Fundamental Theorem of Calculus to evaluate a definite integral.

The Fundamental Theorem of Calculus connects differentiation and integration. It's like a bridge between two worlds. The theorem states that if you have an antiderivative of a function, you can find the definite integral of that function over an interval by evaluating the antiderivative at the endpoints of the interval.

In practical terms, this means:

• Find the antiderivative \( F(x) \) of the function \( f(x) \).
• Calculate \( F(b) - F(a) \), where \( a \) and \( b \) are the lower and upper limits of the integration interval respectively.
• This difference gives the value of the definite integral.

In our example, we found \( F(x) = \frac{x^4}{4} - x^2 + 3x \) and used this to find \( F(3) \) and \( F(-2) \), then calculated \( F(3) - F(-2) \). This process allows us to determine the total area under the curve described by the function from \( x = -2 \) to \( x = 3 \).

A definite integral represents the area under the curve of a function from one point to another on a graph. It's not just about the size, but also about the direction. Positive areas are above the x-axis, while negative areas are below.

To compute a definite integral,

• You first find the antiderivative of the function as seen in previous sections.
• Next, apply the Fundamental Theorem of Calculus.
• Evaluate the antiderivative at the upper limit and subtract the evaluation at the lower limit.

In our task, we dealt with \( \int_{-2}^{3}(x^3 - 2x + 3) \, dx \). We first determined the antiderivative and with the Fundamental Theorem of Calculus, calculated \( F(3) \) and \( F(-2) \). The difference, \( F(3) - F(-2) \), gave us the value \( \frac{105}{4} \). This is the net area between the function \( x^3 - 2x + 3 \) and the x-axis from \( x = -2 \) to \( x = 3 \). This method helps us calculate precise areas, illustrating how intricate calculus can simplify complex geometric concepts.