ental Theorem of Calculus. Remember to include a "+ C" if appropriate.

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Evaluate the definite integral using Part 1 of the Fundamental Theorem of Calculus. Remember to include a "+ C" if appropriate.

### Evaluating a Definite Integral

**Evaluate the definite integral**
$$\int_{0}^{4} (5e^x + 16x) \, dx$$

**Using Part 1 of the Fundamental Theorem of Calculus.**

Enter the exact answer. Remember to include a "+ C" if appropriate.

### Detailed Explanation of the Integral and Concepts

* **Definite Integral**: Evaluates the net area under a curve within a specific interval.
* **Fundamental Theorem of Calculus (Part 1)**: Relates the concept of differentiating a function with the concept of integrating a function. Specifically, it states that if \( F \) is an antiderivative of \( f \) on an interval \([a, b]\), then:
  $$\int_{a}^{b} f(x) \, dx = F(b) - F(a)$$

**Steps to Solve the Integral**:
1. **Find the antiderivative**: Calculate the antiderivative of each term in the integrand.
   
   * Antiderivative of \( 5e^x \) is \( 5e^x \)
   * Antiderivative of \( 16x \) is \( 8x^2 \)
   
   Combining these, we get:

   $$\int (5e^x + 16x) \, dx = 5e^x + 8x^2 + C$$

2. **Evaluate the definite integral using the limits 0 and 4**:
   Substitute the limits into the antiderivative:
   $$\left[ 5e^x + 8x^2 \right]_{0}^{4} = \left( 5e^4 + 8(4^2) \right) - \left( 5e^0 + 8(0^2) \right)$$

3. **Simplify the expression**:
   $$
   = \left( 5e^4 + 8 \times 16 \right) - \left( 5 \times 1 + 8 \times 0 \right)
   $$
   $$
   = 5e^4 + 128 - 5
   $$
   $$
   = 5e^4 + 123
   $$

The exact answer to the integral is \( 5e^4
Transcribed Image Text:### Evaluating a Definite Integral **Evaluate the definite integral** $$\int_{0}^{4} (5e^x + 16x) \, dx$$ **Using Part 1 of the Fundamental Theorem of Calculus.** Enter the exact answer. Remember to include a "+ C" if appropriate. ### Detailed Explanation of the Integral and Concepts * **Definite Integral**: Evaluates the net area under a curve within a specific interval. * **Fundamental Theorem of Calculus (Part 1)**: Relates the concept of differentiating a function with the concept of integrating a function. Specifically, it states that if \( F \) is an antiderivative of \( f \) on an interval \([a, b]\), then: $$\int_{a}^{b} f(x) \, dx = F(b) - F(a)$$ **Steps to Solve the Integral**: 1. **Find the antiderivative**: Calculate the antiderivative of each term in the integrand. * Antiderivative of \( 5e^x \) is \( 5e^x \) * Antiderivative of \( 16x \) is \( 8x^2 \) Combining these, we get: $$\int (5e^x + 16x) \, dx = 5e^x + 8x^2 + C$$ 2. **Evaluate the definite integral using the limits 0 and 4**: Substitute the limits into the antiderivative: $$\left[ 5e^x + 8x^2 \right]_{0}^{4} = \left( 5e^4 + 8(4^2) \right) - \left( 5e^0 + 8(0^2) \right)$$ 3. **Simplify the expression**: $$ = \left( 5e^4 + 8 \times 16 \right) - \left( 5 \times 1 + 8 \times 0 \right) $$ $$ = 5e^4 + 128 - 5 $$ $$ = 5e^4 + 123 $$ The exact answer to the integral is \( 5e^4
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