Find the Antiderivative f(x)=1/(x^8)
The question is asking for the calculation of the antiderivative (also known as the indefinite integral) of the function f(x) = 1/(x^8). In calculus, finding the antiderivative is the process of determining the function F(x) such that its derivative is equal to the given function f(x). The antiderivative involves reversing the process of differentiation, and for this particular function, it requires applying the rules of integration to a power of x with a negative exponent.
$f \left(\right. x \left.\right) = \frac{1}{x^{8}}$
Answer
Solution:
Step 1:
Identify the antiderivative $F(x)$ by integrating the function $f(x)$.
$$F(x) = \int f(x) \, dx$$
Step 2:
Write down the integral that needs to be solved.
$$F(x) = \int \frac{1}{x^8} \, dx$$
Step 3:
Utilize the properties of exponents.
Step 3.1:
Rewrite the integrand using a negative exponent.
$$\int x^{-8} \, dx$$
Step 3.2:
Simplify the expression using exponent rules.
Step 3.2.1:
Apply the rule that $(a^m)^n = a^{mn}$.
$$\int x^{-8} \, dx$$
Step 3.2.2:
Calculate $8 \times (-1)$.
$$\int x^{-8} \, dx$$
Step 4:
Integrate using the Power Rule for integration, which states that $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$.
$$-\frac{1}{7} x^{-7} + C$$
Step 5:
Express the result in a simplified form.
Step 5.1:
Rewrite the expression with a positive exponent.
$$-\frac{1}{7} \cdot \frac{1}{x^7} + C$$
Step 5.2:
Final simplification.
Step 5.2.1:
Combine the constants and the variable's exponent.
$$-\frac{1}{7x^7} + C$$
Step 5.2.2:
Position the constant coefficient appropriately.
$$-\frac{1}{7x^7} + C$$
Step 6:
Conclude with the antiderivative of $f(x) = \frac{1}{x^8}$.
$$F(x) = -\frac{1}{7x^7} + C$$
Knowledge Notes:
Antiderivative: The antiderivative of a function $f(x)$ is another function $F(x)$ such that $F'(x) = f(x)$. It is also known as the indefinite integral and is represented by the integral symbol without bounds.
Integral Setup: To find the antiderivative, we set up an integral of the form $\int f(x) \, dx$.
Exponent Rules: When dealing with exponents, we can use rules like $x^{-n} = \frac{1}{x^n}$ and $(a^m)^n = a^{mn}$ to simplify expressions.
Power Rule for Integration: This rule states that the integral of $x^n$ with respect to $x$ is $\frac{x^{n+1}}{n+1} + C$, provided $n \neq -1$. The constant $C$ represents the constant of integration, which arises because the derivative of a constant is zero.
Simplification: After finding the antiderivative, we often need to rewrite it in a simplified or more conventional form, which may involve expressing negative exponents as fractions or combining constants.
