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


∫₁² (z² + 4) / z dz

Verified step by step guidance

1. Step 1: Simplify the integrand. The given integral is ∫₁² (z² + 4) / z dz. Split the fraction into two terms: (z² / z) + (4 / z). This simplifies to ∫₁² (z + 4/z) dz.
2. Step 2: Break the integral into two separate integrals. Using the linearity property of integrals, rewrite the expression as ∫₁² z dz + ∫₁² (4/z) dz.
3. Step 3: Compute the antiderivative of each term. For the first term, ∫ z dz, the antiderivative is (1/2)z². For the second term, ∫ (4/z) dz, the antiderivative is 4ln|z|.
4. Step 4: Apply the Fundamental Theorem of Calculus. Evaluate the antiderivatives at the upper limit (z = 2) and subtract the evaluation at the lower limit (z = 1). For (1/2)z², compute [(1/2)(2²) - (1/2)(1²)]. For 4ln|z|, compute [4ln(2) - 4ln(1)].
5. Step 5: Combine the results from both terms. Add the results from the evaluations of (1/2)z² and 4ln|z| to obtain the final value of the definite integral.

Key Concepts

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 reflects the accumulation of quantities, such as area, over that interval.

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, 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. Finding an antiderivative is essential for evaluating definite integrals using the Fundamental Theorem of Calculus. For example, if F(z) is an antiderivative of f(z), then the definite integral from a to b can be computed as F(b) - F(a), providing the net area under the curve of f(z) between those limits.