Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus

∫₁⁹ 2/(√𝓍) d𝓍

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Step 1: Recognize that the integral ∫₁⁹ 2/(√𝓍) d𝓍 can be rewritten using exponents. Recall that √𝓍 is equivalent to 𝓍^(1/2), so the integrand becomes 2 * 𝓍^(-1/2).

Step 2: Apply the power rule for integration. The general formula for integrating 𝓍^n is ∫𝓍^n d𝓍 = (𝓍^(n+1))/(n+1) + C, where n ≠ -1. Here, n = -1/2.

Step 3: Compute the antiderivative of 2 * 𝓍^(-1/2). Using the power rule, the antiderivative becomes 2 * (𝓍^(1/2))/(1/2), which simplifies to 4 * 𝓍^(1/2).

Step 4: Use the Fundamental Theorem of Calculus to evaluate the definite integral. Substitute the limits of integration (𝓍 = 1 and 𝓍 = 9) into the antiderivative. This gives [4 * √𝓍] evaluated from 1 to 9.

Step 5: Calculate the difference between the values of the antiderivative at the upper and lower limits. Specifically, compute 4 * √9 - 4 * √1. Simplify the expression to find the final result.

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Definite Integrals

Definite integrals represent the signed area under a curve between two specified limits. They are calculated using the integral symbol with lower and upper bounds, indicating the interval over which the function is evaluated. The result of a definite integral is a numerical value that quantifies this area, which can be interpreted in various contexts, such as physics or economics.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links the concept of differentiation with integration, providing a method to evaluate definite integrals. It states that if a function is continuous on an interval, then the integral of its derivative over that interval equals the difference in the values of the original function at the endpoints. This theorem allows for the computation of definite integrals by finding an antiderivative of the integrand.

Antiderivatives

An antiderivative of a function is another function whose derivative yields the original function. In the context of definite integrals, finding an antiderivative is crucial because it allows us to apply the Fundamental Theorem of Calculus. For example, if we need to evaluate the integral of a function, we first determine its antiderivative and then compute the difference between its values at the upper and lower limits of integration.

Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus∫₁⁹ 2/(√𝓍) d𝓍