2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
3. ∫ (3x)/√(x + 4) dx

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Step 1: Recognize that the integral ∫ (3x)/√(x + 4) dx involves a combination of algebraic and radical expressions. To simplify, consider substitution techniques to reduce the complexity of the integrand.

Step 2: Let u = x + 4. Then, compute the derivative of u with respect to x: du/dx = 1, which implies that du = dx. Also, note that x = u - 4.

Step 3: Substitute u and du into the integral. Replace x with (u - 4) and √(x + 4) with √u. The integral becomes ∫ (3(u - 4))/√u du.

Step 4: Split the integral into two simpler parts: ∫ (3u/√u) du - ∫ (12/√u) du. Simplify each term using the property √u = u^(1/2). For the first term, 3u/√u simplifies to 3u^(1/2). For the second term, 12/√u simplifies to 12u^(-1/2).

Step 5: Integrate each term separately. Use the power rule for integration: ∫ u^n du = (u^(n+1))/(n+1) + C, where n ≠ -1. For the first term, integrate 3u^(1/2). For the second term, integrate -12u^(-1/2). Combine the results and substitute back u = x + 4 to express the solution in terms of x.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Integration Techniques

Integration techniques are methods used to find the integral of a function. Common techniques include substitution, integration by parts, and partial fractions. Understanding these methods is crucial for evaluating more complex integrals, as they allow for simplification and manipulation of the integrand to make integration feasible.

Substitution Method

The substitution method is a technique where a new variable is introduced to simplify the integral. By substituting a part of the integrand with a new variable, the integral can often be transformed into a more manageable form. This method is particularly useful when dealing with composite functions or when the integrand contains a function and its derivative.

Definite vs. Indefinite Integrals

Definite integrals calculate the area under a curve between two specific limits, while indefinite integrals represent a family of functions and include a constant of integration. Understanding the difference is essential for correctly applying integration techniques, as the approach may vary depending on whether the integral is definite or indefinite.

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