2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
35. ∫ x³/√(4x² + 16) dx

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Step 1: Observe the integral ∫ x³/√(4x² + 16) dx. Notice that the denominator involves a square root of a quadratic expression. This suggests that substitution or trigonometric techniques might be useful.

Step 2: Simplify the quadratic expression inside the square root. Factor out the constant 4 from the term 4x² + 16 to rewrite it as 4(x² + 4). This simplifies the square root to √(4(x² + 4)) = 2√(x² + 4).

Step 3: Perform substitution to simplify the integral. Let u = x² + 4, which implies that du = 2x dx. Rewrite the integral in terms of u: ∫ x³/√(4x² + 16) dx = ∫ x² * x / (2√(u)) dx = ∫ x² / (2√(u)) * (du / 2x).

Step 4: Simplify further using the substitution. Since u = x² + 4, x² = u - 4. Substitute this into the integral: ∫ x² / (2√(u)) * (du / 2x) = ∫ (u - 4) / (2√(u)) * (du / 2x). Cancel out x terms and simplify the integral.

Step 5: Break the integral into simpler parts. After substitution and simplification, you will have two separate integrals to evaluate: one involving u/√(u) and another involving constants. Use integration techniques such as power rule or logarithmic integration to solve each part.

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Integration Techniques

Integration techniques are methods used to find the integral of a function. Common techniques include substitution, integration by parts, and trigonometric substitution. Each method is suited for different types of integrals, and understanding when to apply each technique is crucial for solving complex integrals effectively.

Substitution Method

The substitution method involves changing the variable of integration to simplify the integral. This is particularly useful when the integrand contains a composite function. By substituting a new variable, the integral can often be transformed into a more manageable form, making it easier to evaluate.

Rational Functions and Square Roots

Rational functions are ratios of polynomials, and integrals involving square roots often require special techniques. In this case, recognizing the structure of the integrand, such as the presence of √(4x² + 16), can guide the choice of substitution or trigonometric identities to simplify the integral before evaluation.

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.35. ∫ x³/√(4x² + 16) dx