f(x)dx =F(b)- F(a) Please explein in detar L, . To evaluate the integral, first find F(s).) How to get from A to B Let f(s) = 1+s F(s)=s-2s 1/2+C lue of E/h) Ela) Thus there is no need to
Family of Curves
A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.
Euclidean Geometry
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.
Lines and Angles
In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.
![**Evaluate the Integral**
Evaluate the integral \(\int_{1}^{\sqrt{6}} \frac{s^2 + \sqrt{5}}{s^2} \, ds\).
Begin by simplifying the integrand of \(\int_{1}^{\sqrt{6}} \frac{s^2 + \sqrt{5}}{s^2} \, ds\) and expressing it in exponential form.
\[
\frac{s^2 + \sqrt{5}}{s^2} = 1 + s^{-3/2}
\]
According to the second part of the fundamental theorem of calculus, if \( f \) is continuous at every point of \([a,b]\) and \( F \) is any antiderivative of \( f \) on \([a,b]\), then the definite integral can be evaluated using the following formula.
\[
\int_{a}^{b} f(x) \, dx = F(b) - F(a)
\]
**Explanation of the Method:**
1. **Simplify the Function:**
- Let \( f(s) = 1 + s^{-3/2} \).
- To evaluate the integral, first find \( F(s) \).
2. **Find \( F(s) \):**
- \( F(s) = s - 2s^{-1/2} + C \)
Because of the subtraction, a constant in \( F(x) \) will not affect the value of \( F(b) - F(a) \). Thus, there is no need to include the constant \( C \).
3. **Integral Evaluation:**
\[
\int_{1}^{\sqrt{6}} \frac{s^2 + \sqrt{5}}{s^2} \, ds = F(\sqrt{6}) - F(1)
\]
\[
= [s - 2s^{-1/2}]_{1}^{\sqrt{6}}
\]
4. **Simplify \( F(\sqrt{6}) \):**
\[
F(\sqrt{6}) = (\sqrt{6}) - 2 \cdot (\sqrt{6})^{-1/2}
\]
\[
= (\sqrt{6}) - 2 \cdot (6^{1/2})^{-1/2}
\]
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