Find the Antiderivative f(x)=12/( square root of 1-x^2)
The given problem is asking for the antiderivative, also known as the indefinite integral, of the function f(x) = 12/(√(1-x^2)). An antiderivative of a function is another function whose derivative gives the original function. The process of finding the antiderivative is known as integration. The specific integral in question involves a rational function where the denominator is the square root of (1-x^2), which is a common form seen when integrating functions using trigonometric substitution, particularly with sine or cosine functions due to their relationship with the unit circle and the Pythagorean identity.
$f \left(\right. x \left.\right) = \frac{12}{\sqrt{1 - x^{2}}}$
Answer
Solution:
Step 1:
Identify the antiderivative $F(x)$ by integrating the given function $f(x)$.
$$F(x) = \int f(x) \, dx$$
Step 2:
Write down the integral that needs to be solved.
$$F(x) = \int \frac{12}{\sqrt{1 - x^2}} \, dx$$
Step 3:
Extract the constant $12$ from the integral as it does not depend on $x$.
$$12 \int \frac{1}{\sqrt{1 - x^2}} \, dx$$
Step 4:
Express the number $1$ as a square, $1^2$, to match the form of a trigonometric identity.
$$12 \int \frac{1}{\sqrt{1^2 - x^2}} \, dx$$
Step 5:
Recognize the integral as the inverse sine function, $\arcsin(x)$.
$$12 (\arcsin(x) + C)$$
Step 6:
Combine the constant multiple with the antiderivative.
$$12 \arcsin(x) + C$$
Step 7:
Conclude with the antiderivative of the function $f(x) = \frac{12}{\sqrt{1 - x^2}}$.
$$F(x) = 12 \arcsin(x) + C$$
Knowledge Notes:
The process of finding the antiderivative involves integrating the given function. The integral of a function $f(x)$ is denoted by $\int f(x) \, dx$, and it represents the accumulation of the area under the curve of $f(x)$.
In this particular problem, we are dealing with the integral of a function that resembles the derivative of the arcsine function. The general form of the integral that corresponds to the arcsine function is $\int \frac{1}{\sqrt{a^2 - x^2}} \, dx = \arcsin\left(\frac{x}{a}\right) + C$, where $C$ is the constant of integration.
The constant multiple rule in integration states that if $k$ is a constant and $f(x)$ is an integrable function, then $\int k \cdot f(x) \, dx = k \cdot \int f(x) \, dx$. This rule allows us to move constants outside the integral.
The antiderivative found through this process is not unique; it includes an arbitrary constant $C$, because the derivative of a constant is zero. This constant represents the family of all antiderivatives of the function.
The inverse trigonometric function $\arcsin(x)$ is the antiderivative of $\frac{1}{\sqrt{1 - x^2}}$ within the domain $-1 \leq x \leq 1$. It is important to recognize this integral form to solve the problem efficiently.
