Evaluate the integrals in Exercises 53–76.
61. ∫(from 0 to 2)dt/√(8+2t²)

Verified step by step guidance

1. Identify the integral to evaluate: \(\int_0^2 \frac{dt}{\sqrt{8 + 2t^2}}\).
2. Factor out the constant inside the square root to simplify the integrand: rewrite \(\sqrt{8 + 2t^2}\) as \(\sqrt{2(4 + t^2)} = \sqrt{2} \sqrt{4 + t^2}\).
3. Rewrite the integral using this simplification: \(\int_0^2 \frac{dt}{\sqrt{2} \sqrt{4 + t^2}} = \frac{1}{\sqrt{2}} \int_0^2 \frac{dt}{\sqrt{4 + t^2}}\).
4. Recognize that the integral \(\int \frac{dt}{\sqrt{a^2 + t^2}}\) has a standard antiderivative: \(\ln|t + \sqrt{t^2 + a^2}| + C\), where \(a\) is a constant.
5. Apply the antiderivative formula with \(a = 2\), evaluate the resulting expression at the limits \(t=2\) and \(t=0\), and subtract to find the definite integral value.

Key Concepts

Definite Integrals

A definite integral calculates the net area under a curve between two specified limits. It involves evaluating the integral function at the upper and lower bounds and subtracting these values. This concept is essential for solving integrals with given limits, such as from 0 to 2.

Integration of Functions Involving Square Roots

Integrals containing square roots often require substitution or recognizing standard integral forms. For example, expressions like ∫ dt/√(a + bt²) can be solved using trigonometric substitution or by identifying it as a form related to inverse hyperbolic functions.

Substitution Method in Integration

The substitution method simplifies integrals by changing variables to transform the integral into a more manageable form. This technique is particularly useful when the integrand contains composite functions, such as expressions inside a square root, enabling easier integration.