Variations on the substitution method Evaluate the following integrals.

∫ 𝓍/(√𝓍―4) d𝓍

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Step 1: Identify the substitution method. Let u = √𝓍 - 4. This substitution simplifies the square root term in the denominator.

Step 2: Differentiate u with respect to 𝓍 to find du. Since u = √𝓍 - 4, differentiate both sides to get du = (1/(2√𝓍)) d𝓍.

Step 3: Rewrite the integral in terms of u. Substitute √𝓍 = u + 4 and d𝓍 = 2√𝓍 du into the integral. This transforms the integral into a simpler form.

Step 4: Simplify the integral. Replace √𝓍 in the numerator with u + 4, and simplify the expression to make it easier to integrate.

Step 5: Integrate the simplified expression with respect to u. After integration, substitute back u = √𝓍 - 4 to return to the original variable 𝓍.

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Key Concepts

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

Substitution Method

The substitution method is a technique used in integration to simplify the process by changing the variable of integration. This involves selecting a new variable, often denoted as 'u', which is a function of the original variable. By substituting 'u' into the integral, the integrand can often be transformed into a simpler form, making it easier to evaluate the integral.

Definite vs. Indefinite Integrals

Integrals can be classified as definite or indefinite. An indefinite integral represents a family of functions and includes a constant of integration, while a definite integral computes the area under the curve between two specified limits. Understanding the difference is crucial for applying the correct evaluation techniques and interpreting the results accurately.

Rational Functions and Their Integration

Rational functions are ratios of polynomials, and their integration often requires specific techniques, such as partial fraction decomposition or substitution. In the given integral, the presence of a square root in the denominator suggests that a substitution may simplify the expression, allowing for easier integration. Recognizing the form of the rational function is key to selecting the appropriate method.

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