Use a substitution of the form u = a𝓍 + b to evaluate the following indefinite integrals.
∫(𝓍 + 1)¹² d𝓍

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Step 1: Identify the substitution. Let u = 𝓍 + 1. This substitution simplifies the expression inside the integral.

Step 2: Compute the derivative of u with respect to 𝓍. Since u = 𝓍 + 1, we have du/d𝓍 = 1, or equivalently, du = d𝓍.

Step 3: Rewrite the integral in terms of u. Substituting u = 𝓍 + 1 and du = d𝓍, the integral becomes ∫u¹² du.

Step 4: Apply the power rule for integration. The integral of uⁿ with respect to u is (uⁿ⁺¹)/(n+1) + C, where C is the constant of integration.

Step 5: Substitute back u = 𝓍 + 1 into the result to express the solution in terms of 𝓍. This completes the evaluation of the indefinite integral.

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Substitution Method

The substitution method is a technique used in integration to simplify the process by changing the variable of integration. By substituting a new variable, often denoted as 'u', for a function of 'x', the integral can be transformed into a more manageable form. This method is particularly useful when dealing with composite functions or when the integrand can be expressed in terms of a simpler variable.

Indefinite Integrals

Indefinite integrals represent a family of functions whose derivative gives the integrand. They are expressed without limits of integration and include a constant of integration, typically denoted as 'C'. The process of finding an indefinite integral involves determining the antiderivative of the function, which can often be achieved through various techniques, including substitution.

Polynomial Functions

Polynomial functions are expressions that consist of variables raised to non-negative integer powers, combined using addition, subtraction, and multiplication. In the context of integration, recognizing polynomial forms is crucial, as they can be integrated using straightforward rules. For example, the integral of x^n is (x^(n+1))/(n+1) + C, which simplifies the evaluation of integrals involving polynomials.

Use a substitution of the form u = a𝓍 + b to evaluate the following indefinite integrals.∫(𝓍 + 1)¹² d𝓍