Textbook Question
Variations on the substitution method Evaluate the following integrals.
∫ y²/(y + 1)⁴ dy
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7. Antiderivatives & Indefinite Integrals
Indefinite Integrals
3:45 minutesProblem 8.R.40
Textbook Question
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
40. ∫ (x² - 4)/(x + 4) dx
Verified step by step guidance1
Step 1: Begin by performing polynomial long division to simplify the integrand. Divide \(x^2 - 4\) by \(x + 4\). This will help rewrite the integral in a simpler form.
Step 2: After performing the division, express the result as \(q(x) + \frac{r(x)}{x+4}\), where \(q(x)\) is the quotient and \(r(x)\) is the remainder. Substitute this into the integral.
Step 3: Split the integral into two parts: \(\int q(x) dx\) and \(\int \frac{r(x)}{x+4} dx\). Evaluate each part separately.
Step 4: For \(\int q(x) dx\), integrate the polynomial term directly using the power rule: \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\).
Step 5: For \(\int \frac{r(x)}{x+4} dx\), use substitution if necessary. Let \(u = x+4\), then \(du = dx\). Rewrite the integral in terms of \(u\) and solve.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration Techniques
Integration techniques are methods used to find the integral of a function. Common techniques include substitution, integration by parts, and partial fraction decomposition. Understanding these methods is crucial for evaluating more complex integrals, as they allow for simplification and manipulation of the integrand to make integration feasible.
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Polynomial Long Division
Polynomial long division is a method used to divide a polynomial by another polynomial of equal or lower degree. In the context of integration, this technique is particularly useful when the degree of the numerator is greater than or equal to the degree of the denominator. By simplifying the integrand through division, the integral can often be expressed in a more manageable form.
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Definite vs. Indefinite Integrals
Definite integrals calculate the area under a curve between two specified limits, while indefinite integrals represent a family of functions and include a constant of integration. Understanding the difference is essential for correctly interpreting the results of integration problems. In this case, the integral presented is indefinite, meaning the solution will include an arbitrary constant.
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