76-81. Table of integrals Use a table of integrals to evaluate the following integrals.
76. ∫ x(2x + 3)⁵ dx

Verified step by step guidance

1. Step 1: Recognize that the integral ∫ x(2x + 3)⁵ dx involves a product of a polynomial term x and a composite function (2x + 3) raised to a power. This suggests using a substitution method to simplify the integral.
2. Step 2: Let u = 2x + 3. Then, compute the derivative of u with respect to x: du/dx = 2, or equivalently, du = 2 dx. Rewrite dx in terms of du: dx = du/2.
3. Step 3: Substitute u and dx into the integral. Replace (2x + 3) with u and dx with du/2. Also, express x in terms of u using the substitution u = 2x + 3, which gives x = (u - 3)/2.
4. Step 4: After substitution, the integral becomes ∫ [(u - 3)/2] * u⁵ * (du/2). Simplify the constants and expand the expression to separate terms for easier integration.
5. Step 5: Use the table of integrals to evaluate the resulting integral term by term. For example, ∫ u⁶ du and ∫ u⁵ du can be directly integrated using standard formulas from the table of integrals. Combine the results and back-substitute u = 2x + 3 to express the solution in terms of x.

Key Concepts

Integration

Integration is a fundamental concept in calculus that involves finding the antiderivative of a function. It is the process of calculating the area under a curve represented by a function over a specified interval. Understanding integration is crucial for evaluating integrals, as it allows us to reverse the process of differentiation.

Table of Integrals

A table of integrals is a reference tool that lists common integrals and their corresponding antiderivatives. It simplifies the process of integration by providing ready-made solutions for frequently encountered functions, allowing students to quickly find the integral of complex expressions without deriving them from first principles.

Polynomial Expansion

Polynomial expansion involves rewriting a polynomial expression in a more manageable form, often using the binomial theorem or distributive property. In the context of the given integral, expanding the expression x(2x + 3)⁵ will make it easier to integrate term by term, as it transforms the integral into a sum of simpler polynomial terms.