Variations on the substitution method Evaluate the following integrals.                                                                                                        
                                                                                                                                                                    
 ∫ y²/(y + 1)⁴ dy

Verified step by step guidance

1. Step 1: Identify the substitution method. Let u = y + 1, which simplifies the denominator. Then, du = dy.
2. Step 2: Rewrite y in terms of u. Since u = y + 1, we have y = u - 1.
3. Step 3: Substitute into the integral. Replace y² with (u - 1)² and (y + 1)⁴ with u⁴, and dy with du. The integral becomes ∫ (u - 1)² / u⁴ du.
4. Step 4: Expand (u - 1)² to simplify the numerator. This gives (u - 1)² = u² - 2u + 1. Substitute this into the integral, resulting in ∫ (u² - 2u + 1) / u⁴ du.
5. Step 5: Break the integral into separate terms. Divide each term in the numerator by u⁴: ∫ (u²/u⁴ - 2u/u⁴ + 1/u⁴) du, which simplifies to ∫ (1/u² - 2/u³ + 1/u⁴) du. Now, integrate each term individually.

Key Concepts

Substitution Method

The substitution method is a technique used in integration to simplify the process by changing the variable of integration. 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.

Integration by Parts

Integration by parts is a technique derived from the product rule of differentiation, used to integrate products of functions. It states that the integral of a product of two functions can be expressed as the product of one function and the integral of the other, minus the integral of the derivative of the first function times the second. This method is helpful when the integrand is a product that can be simplified through differentiation.

Polynomial Long Division

Polynomial long division is a method used to divide a polynomial by another polynomial of lower degree. This technique is particularly useful when the integrand is a rational function, allowing the integrand to be expressed as a sum of a polynomial and a proper fraction. By simplifying the integrand in this way, it becomes easier to integrate each term separately, facilitating the overall integration process.