Change of variables Use the change of variables u³ = 𝓍² ― 1 to evaluate the integral ∫₁³ 𝓍∛(𝓍²―1) d𝓍 .

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Step 1: Identify the substitution. Let u³ = 𝓍² - 1. Differentiate both sides with respect to 𝓍 to find the relationship between du and d𝓍. Differentiating gives 3u² du = 2𝓍 d𝓍.

Step 2: Solve for d𝓍 in terms of u and du. Rearrange the equation to get d𝓍 = (3u²)/(2𝓍) du.

Step 3: Rewrite 𝓍 in terms of u using the substitution u³ = 𝓍² - 1. Solving for 𝓍 gives 𝓍 = √(u³ + 1).

Step 4: Change the limits of integration. When 𝓍 = 1, u³ = 1² - 1 = 0, so u = 0. When 𝓍 = 3, u³ = 3² - 1 = 8, so u = 2.

Step 5: Substitute everything into the integral. Replace 𝓍, d𝓍, and the integrand with their expressions in terms of u. The integral becomes ∫₀² √(u³ + 1) ∛(u³) * (3u²)/(2√(u³ + 1)) du. Simplify the integrand and proceed to evaluate the integral.

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Change of Variables

The change of variables technique in calculus allows us to simplify integrals by substituting a new variable for the original variable. This method can transform a complex integral into a more manageable form, making it easier to evaluate. The substitution must be accompanied by the appropriate adjustment of the differential, ensuring that the limits of integration and the integrand are correctly modified.

Definite Integrals

A definite integral represents the signed area under a curve between two specified limits. It is denoted as ∫_a^b f(x) dx, where 'a' and 'b' are the lower and upper limits, respectively. Evaluating a definite integral involves finding the antiderivative of the function and then applying the Fundamental Theorem of Calculus to compute the difference between the values at the limits.

Integration by Substitution

Integration by substitution is a method used to simplify the process of integration by changing the variable of integration. This technique often involves identifying a part of the integrand that can be replaced with a single variable, which simplifies the integral. The derivative of the substituted variable must also be accounted for, ensuring that the integral remains equivalent to the original.