23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.

∫ (eˣ⁺²) dx

검증된 단계별 안내

Step 1: Recognize that the integral ∫(e^(x+2)) dx involves an exponential function. The general rule for integrating e^(u) is ∫e^(u) du = e^(u) + C, where u is a function of x.

Step 2: Identify the exponent in the given integral. Here, u = x + 2. To apply the rule, we need to check if the derivative of u with respect to x is present in the integral.

Step 3: Compute the derivative of u = x + 2 with respect to x. The derivative is du/dx = 1. Since the derivative is 1, we can directly integrate without additional adjustments.

Step 4: Apply the integration rule for e^(u). The integral becomes e^(x+2) + C, where C is the constant of integration.

Step 5: Verify your result by differentiating e^(x+2) + C. The derivative of e^(x+2) is e^(x+2), and the derivative of the constant C is 0. This confirms that the original integrand is recovered, ensuring the solution is correct.

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주요 개념

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Indefinite Integrals

Indefinite integrals represent a family of functions whose derivative is the integrand. They are expressed without limits and include a constant of integration, typically denoted as 'C'. The process of finding an indefinite integral is known as integration, which is the reverse operation of differentiation.

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = a * e^(bx), where 'e' is the base of natural logarithms, approximately equal to 2.71828. In the context of integration, the integral of e^(kx) is (1/k)e^(kx) + C, which is crucial for solving integrals involving exponential terms.

Differentiation Check

Checking work by differentiation involves taking the derivative of the obtained integral to verify that it matches the original integrand. This process ensures that the integration was performed correctly and helps identify any potential errors in the integration process.

23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.∫ (eˣ⁺²) dx