Find the Antiderivative g(x)=1/(x^5)
The question asks for the calculation of the antiderivative, also commonly known as the indefinite integral, of a given function g(x) which is equal to 1/(x^5). This involves finding a function F(x) whose derivative is g(x). The antiderivative includes an arbitrary constant of integration, usually denoted as C, because the derivative of a constant is zero. This question is asking you to apply integration techniques to find the function F(x) that satisfies F'(x) = g(x).
$g \left(\right. x \left.\right) = \frac{1}{x^{5}}$
Answer
Solution:
Step 1:
Identify the antiderivative $G(x)$ by integrating the function $g(x)$.
$$G(x) = \int g(x) \, dx$$
Step 2:
Write down the integral that needs to be solved.
$$G(x) = \int \frac{1}{x^5} \, dx$$
Step 3:
Utilize the rules for manipulating exponents.
Step 3.1:
Rewrite the integrand by using negative exponents.
$$\int x^{-5} \, dx$$
Step 3.2:
Simplify the expression by applying exponent multiplication rules.
Step 3.2.1:
Invoke the rule $(a^m)^n = a^{mn}$ for exponents.
$$\int x^{-5} \, dx$$
Step 3.2.2:
Perform the multiplication of the exponents.
$$\int x^{-5} \, dx$$
Step 4:
Apply the Power Rule for integration, $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ when $n \neq -1$.
$$-\frac{1}{4}x^{-4} + C$$
Step 5:
Express the result in a more simplified form.
Step 5.1:
Convert the negative exponent to a denominator.
$$-\frac{1}{4} \cdot \frac{1}{x^4} + C$$
Step 5.2:
Final simplification.
Step 5.2.1:
Combine the constants.
$$-\frac{1}{4x^4} + C$$
Step 5.2.2:
Ensure the constant is properly positioned.
$$-\frac{1}{4x^4} + C$$
Step 6:
Present the final antiderivative of $g(x) = \frac{1}{x^5}$.
$$G(x) = -\frac{1}{4x^4} + C$$
Knowledge Notes:
To solve for the antiderivative of a function $g(x) = \frac{1}{x^5}$, we follow a series of steps that involve understanding and applying integral calculus concepts.
Indefinite Integral: The antiderivative of a function $g(x)$ is found by calculating the indefinite integral of $g(x)$ with respect to $x$.
Exponent Rules: When dealing with exponents in integrals, we can use the rule that $x^{-n} = \frac{1}{x^n}$ to rewrite the integrand in a form that is easier to integrate.
Power Rule for Integration: This is a fundamental rule in calculus that states that the integral of $x^n$ with respect to $x$ is $\frac{x^{n+1}}{n+1}$, plus a constant of integration $C$, as long as $n$ is not equal to $-1$.
Simplification: After applying the Power Rule, it's important to simplify the expression to its most reduced form. This may involve rewriting negative exponents as denominators and combining constants.
Constant of Integration: When finding the indefinite integral, we always add a constant of integration $C$ because the antiderivative is not unique; adding any constant to a function does not change its derivative.
By following these steps and applying these rules, we can find the antiderivative of the given function.
