7–64. Integration review Evaluate the following integrals.
57. ∫ dx / (x¹⸍² + x³⸍²)

Verified step by step guidance

1. Step 1: Recognize that the integral ∫ dx / (x^(1/2) + x^(3/2)) involves fractional exponents. Rewrite the denominator in terms of a common factor to simplify the expression. Factor out x^(1/2) from the denominator: x^(1/2) + x^(3/2) = x^(1/2)(1 + x).
2. Step 2: Substitute the simplified denominator into the integral. The integral becomes ∫ dx / (x^(1/2)(1 + x)).
3. Step 3: Perform a substitution to simplify the integral further. Let u = 1 + x, which implies du = dx. Also, note that x = u - 1, and x^(1/2) = (u - 1)^(1/2). Rewrite the integral in terms of u.
4. Step 4: After substitution, the integral becomes ∫ du / ((u - 1)^(1/2) * u). Break this into manageable parts or consider partial fraction decomposition if applicable.
5. Step 5: Solve the resulting integral step by step, applying appropriate integration techniques such as substitution, partial fractions, or standard integral formulas. Combine the results and simplify.

Key Concepts

Integration Techniques

Integration techniques are methods used to evaluate integrals, which can include substitution, integration by parts, and partial fraction decomposition. Understanding these techniques is essential for solving complex integrals, especially when dealing with rational functions or expressions that can be simplified.

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. In the given integral, the expression in the denominator is a polynomial, and recognizing its structure is crucial for determining the appropriate method for integration, such as factoring or simplifying the expression.

Limits of Integration

Limits of integration define the interval over which the integral is evaluated. While the given integral is indefinite, understanding how to apply limits is important for definite integrals, as it affects the final value of the integral and the interpretation of the area under the curve.