Find the Antiderivative f(x)=8/(x^5)

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 $x^{-5}$.

$8(-\frac{1}{4}{x}^{-4}+C)$

Step 6:

Proceed to simplify the expression.

Step 6.1:

Combine the constant and the variable with the negative exponent.

$8(-\frac{x^{-4}}{4}+C)$

Step 6.2:

Simplify the expression by distributing the $8$ and combining terms.

$-2\frac{8}{4x^4}+C$

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:

1. Understanding the Problem: Recognize that finding an antiderivative means integrating the given function.

2. Setting Up the Integral: Write the integral of the function, which represents the antiderivative.

3. Constant Multiple Rule: If a constant is multiplied by a function, it can be pulled out of the integral.

4. Exponent Rules: When dealing with exponents in the denominator, we can rewrite them as negative exponents in the numerator.

5. Power Rule for Integration: The power rule states that $\int x^{n}dx=\frac{x^{n+1}}{n+1}+C$, where $n\ne -1$ and $C$ is the constant of integration.

6. Simplification: After integrating, simplify the expression by combining like terms, reducing fractions, and applying exponent rules.

7. 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.