Problem 56 - Calculus Single Variable

Question

\(F(x)\) is a function of a variable \(x\) that appears in a limit (or in the limits) of integration of a given definite integral. Express \(F(x)\) explicitly by calculating the integral. $$ F(x)=\int_{x}^{2 x}\left(2 t+3 t^{2}\right) d t $$

Step-by-Step Solution

Answer

The function \(F(x)\) is explicitly \(3x^2 + 7x^3\).

Step 1: Define the Integral Limits

The function given is \(F(x) = \int_{x}^{2x}(2t + 3t^2) \, dt\). This means that the function is defined by an integral with variable limits from \(x\) to \(2x\). We will integrate with respect to \(t\) over this interval.

Step 2: Find the Antiderivative

To find the antiderivative of the integrand \(2t + 3t^2\), integrate each term separately. The antiderivative of \(2t\) is \(t^2\) and the antiderivative of \(3t^2\) is \(t^3\). Therefore, the antiderivative of \(2t + 3t^2\) is \(t^2 + t^3\).

Step 3: Evaluate the Antiderivative at the Upper Limit

Substitute the upper limit \(2x\) into the antiderivative, giving \((2x)^2 + (2x)^3 = 4x^2 + 8x^3\).

Step 4: Evaluate the Antiderivative at the Lower Limit

Substitute the lower limit \(x\) into the antiderivative, resulting in \(x^2 + x^3\).

Step 5: Compute the Definite Integral

Subtract the value obtained by substituting the lower limit from the value obtained by substituting the upper limit: \( (4x^2 + 8x^3) - (x^2 + x^3) \). This simplifies to \(3x^2 + 7x^3\).

Step 6: Express the Function Explicitly

Thus, the function \(F(x)\) can be explicitly expressed as \(F(x) = 3x^2 + 7x^3\).

Key Concepts

Antiderivative

When dealing with integrals, the antiderivative is a crucial concept. Essentially, it's the reverse process of differentiation. If we have a function, finding its antiderivative means determining a new function whose derivative is the original one. Consider the integral function \[ \int(2t + 3t^2) \, dt \] Here, the task is to break it down into simpler parts:

• The antiderivative of \(2t\) is \(t^2\).
• The antiderivative of \(3t^2\) is \(t^3\).

Add these components together, and the antiderivative becomes \(t^2 + t^3\). This function is what you derive from the original integrand, using basic integration rules.

Variable Limits of Integration

In definite integrals, limits determine the range of the variable over which integration takes place. Usually, these limits are constants, allowing a simple computation. However, when they are variables themselves—like in the function:\[ F(x) = \int_{x}^{2x}(2t + 3t^2) \, dt \]things become a bit more interesting. Here, the lower limit is \(x\) and the upper limit is \(2x\). This setup means the limits are not fixed numbers, but rather expressions that include the variable \(x\). This variation directly affects the computation outcome and calls for a careful substitution of these variable limits within the antiderivative to compute the integral accurately.

Evaluation of Integrals

With all integrals, especially definite integrals, evaluating them properly is the final step. Given an antiderivative, like our \(t^2 + t^3\), determining the definite integral involves substituting the variable limits into the antiderivative.
To start, plug in the upper limit \(2x\) into the antiderivative, resulting in:

• \((2x)^2 + (2x)^3 = 4x^2 + 8x^3\)

Next, substitute the lower limit \(x\):

• \(x^2 + x^3\)

Finally, subtract the lower limit evaluation from the upper limit evaluation:
\[ (4x^2 + 8x^3) - (x^2 + x^3) = 3x^2 + 7x^3 \]This subtraction gives us a clean, explicit expression of the integral. Hence, the fully evaluated integral is \(3x^2 + 7x^3\), explaining the behavior and magnitude of the function \(F(x)\) across the range specified.