Find the Antiderivative f(x)=8/(x^5)
The problem requires determining the antiderivative (also known as the indefinite integral) of a given function f(x) = 8/(x^5). To solve this, one needs to apply the techniques of integration to find a function F(x), such that the derivative of F(x) with respect to x is equal to the original function f(x). The process will likely involve using the power rule for integration on the given function, where the exponent is negative, causing the need to adjust the integral accordingly.
$f \left(\right. x \left.\right) = \frac{8}{x^{5}}$
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.
$F(x) = \int \frac{8}{x^5} \, dx$
Step 3:
Extract the constant $8$ from the integral as it does not depend on $x$.
$8 \int \frac{1}{x^5} \, dx$
Step 4:
Utilize the properties of exponents to rewrite the integrand.
Step 4.1:
Express $x^5$ in the denominator as $x$ raised to the power of $-5$.
$8 \int x^{-5} \, dx$
Step 5:
Apply the Power Rule for integration to find the integral of $x^{-5}$.
$8 \left( -\frac{1}{4} x^{-4} + C \right)$
Step 6:
Proceed to simplify the expression.
Step 6.1:
Combine the constant and the variable with the negative exponent.
$8 \left( -\frac{x^{-4}}{4} + C \right)$
Step 6.2:
Simplify the expression by distributing the $8$ and combining terms.
$-2 \frac{8}{4 x^4} + C$
Step 6.3:
Reduce the fraction by dividing the numerator and the denominator by their greatest common factor.
$-\frac{2}{x^4} + C$
Step 7:
Conclude with the antiderivative of the function $f(x) = \frac{8}{x^5}$.
$F(x) = -\frac{2}{x^4} + C$
Knowledge Notes:
To solve for the antiderivative of a function, we follow a systematic approach:
Understanding the Problem: Recognize that finding an antiderivative means integrating the given function.
Setting Up the Integral: Write the integral of the function, which represents the antiderivative.
Constant Multiple Rule: If a constant is multiplied by a function, it can be pulled out of the integral.
Exponent Rules: When dealing with exponents in the denominator, we can rewrite them as negative exponents in the numerator.
Power Rule for Integration: The power rule states that $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$, where $n \neq -1$ and $C$ is the constant of integration.
Simplification: After integrating, simplify the expression by combining like terms, reducing fractions, and applying exponent rules.
Final Answer: The final simplified expression represents the antiderivative of the original function.
In the given problem, we applied these steps to integrate $f(x) = \frac{8}{x^5}$, which involved pulling out the constant, applying the power rule, and simplifying the resulting expression.
