1. Let's start by stating the problem: You want to know how to decide which integration technique to use, such as partial fractions or others, and how the power of terms influences this decision. 2. When calculating integrals, the choice of technique depends on the form of the integrand (the function to be integrated). Common techniques include substitution, integration by parts, partial fractions, trigonometric substitution, and recognizing standard forms. 3. One important rule is to look at the algebraic structure of the integrand. For example, if the integrand is a rational function (a ratio of polynomials), partial fraction decomposition is often useful, especially when the degree of the numerator is less than the degree of the denominator. 4. The "power" or degree of polynomials plays a key role: - If the numerator's degree is greater than or equal to the denominator's degree, perform polynomial division first to simplify. - If the denominator factors into linear or quadratic factors, partial fractions can be applied. 5. For powers of expressions, such as $(x^2 + 1)^n$, the value of $n$ guides the method: - If $n$ is a positive integer, substitution or binomial expansion might help. - If $n$ is negative, partial fractions or trigonometric substitution might be better. 6. Other considerations: - If the integrand contains products of functions, integration by parts may be suitable. - If the integrand involves roots or trigonometric expressions, trigonometric substitution or special substitutions are useful. 7. Summary: To decide the technique, analyze the integrand's form, factor polynomials if rational, check degrees (powers), and identify patterns matching known integral types. This approach helps you choose the most efficient integration method.