The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀² (s + 1) / √(4 − s²) ds

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Identify the integral to be solved: \(\int_0^2 \frac{s + 1}{\sqrt{4 - s^2}} \, ds\).

Split the integral into two separate integrals for easier handling: \(\int_0^2 \frac{s}{\sqrt{4 - s^2}} \, ds + \int_0^2 \frac{1}{\sqrt{4 - s^2}} \, ds\).

For the first integral \(\int_0^2 \frac{s}{\sqrt{4 - s^2}} \, ds\), use the substitution method. Let \(u = 4 - s^2\), then find \(du\) and express \(s \, ds\) in terms of \(du\).

For the second integral \(\int_0^2 \frac{1}{\sqrt{4 - s^2}} \, ds\), recognize it as a standard integral form related to the arcsine function: \(\int \frac{1}{\sqrt{a^2 - x^2}} \, dx = \arcsin\left(\frac{x}{a}\right) + C\).

Evaluate both integrals using the substitution and standard integral results, then combine the results and apply the limits from 0 to 2 to find the value of the original integral.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Definite Integrals

A definite integral calculates the net area under a curve between two limits. It is evaluated by finding the antiderivative of the integrand and then applying the Fundamental Theorem of Calculus to compute the difference at the upper and lower bounds.

Substitution Method

The substitution method simplifies integrals by changing variables to transform the integral into a more manageable form. It often involves setting a part of the integrand equal to a new variable, which helps in integrating complex expressions like those involving square roots.

Integrals Involving Square Roots of Quadratic Expressions

Integrals with terms like √(a² − x²) often require trigonometric substitution or recognizing standard integral forms. These techniques help convert the integral into a trigonometric integral, which is easier to evaluate without tables.

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