Find the Antiderivative f(x)=-1/7x^(-8/7)
The problem is asking for the calculation of an antiderivative (also known as an indefinite integral) of the function f(x) = -1/7x^(-8/7). This involves finding a function F(x) whose derivative with respect to x is equal to the given function f(x). Essentially, you are being tasked with performing the reverse operation of differentiation on the given function. The antiderivative should include a constant of integration, since when taking the derivative of a constant, the result is zero, which means that any constant could have been present in the original antiderivative.
$f \left(\right. x \left.\right) = - \frac{1}{7} x^{- \frac{8}{7}}$
Answer
Solution:
Step:1
To determine the antiderivative $F(x)$, we integrate the given function $f(x)$.
$$F(x) = \int f(x) \, dx$$
Step:2
Prepare the integral for computation.
$$F(x) = \int -\frac{1}{7} x^{-\frac{8}{7}} \, dx$$
Step:3
Extract the constant $-\frac{1}{7}$ from the integral as it does not depend on $x$.
$$-\frac{1}{7} \int x^{-\frac{8}{7}} \, dx$$
Step:4
Apply the Power Rule for integration to find the integral of $x^{-\frac{8}{7}}$.
$$-\frac{1}{7} \left( -7x^{-\frac{1}{7}} + C \right)$$
Step:5
Proceed to simplify the expression.
Step:5.1
Express $-\frac{1}{7} \left( -7x^{-\frac{1}{7}} + C \right)$ as $-\frac{1}{7} \left( -7 \frac{1}{x^{\frac{1}{7}}} \right) + C$.
$$-\frac{1}{7} \left( -7 \frac{1}{x^{\frac{1}{7}}} \right) + C$$
Step:5.2
Begin simplification.
Step:5.2.1
Combine the constants $-7$ and $\frac{1}{x^{\frac{1}{7}}}$.
$$-\frac{1}{7} \cdot \frac{-7}{x^{\frac{1}{7}}} + C$$
Step:5.2.2
Position the negative sign outside the fraction.
$$-\frac{1}{7} \left( -\frac{7}{x^{\frac{1}{7}}} \right) + C$$
Step:5.2.3
Multiply $-1$ by $-1$.
$$1 \left( \frac{1}{7} \right) \frac{7}{x^{\frac{1}{7}}} + C$$
Step:5.2.4
Multiply $\frac{1}{7}$ by $1$.
$$\frac{1}{7} \cdot \frac{7}{x^{\frac{1}{7}}} + C$$
Step:5.2.5
Multiply $\frac{1}{7}$ by $\frac{7}{x^{\frac{1}{7}}}$.
$$\frac{7}{7x^{\frac{1}{7}}} + C$$
Step:5.2.6
Eliminate the common factors.
$$\frac{\cancel{7}}{\cancel{7}x^{\frac{1}{7}}} + C$$
Step:5.2.7
Finalize the simplified expression.
$$\frac{1}{x^{\frac{1}{7}}} + C$$
Step:6
Conclude with the antiderivative of the function $f(x) = -\frac{1}{7} x^{-\frac{8}{7}}$.
$$F(x) = \frac{1}{x^{\frac{1}{7}}} + C$$
Knowledge Notes:
To solve such a problem, one must be familiar with several key concepts:
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.
Indefinite Integral: The process of finding the antiderivative is called integration. The indefinite integral of a function $f(x)$ with respect to $x$ is denoted by $\int f(x) \, dx$ and represents a family of functions that differ by a constant $C$.
Constant Multiple Rule: When integrating a constant multiplied by a function, the constant can be factored out of the integral. For example, $\int k \cdot f(x) \, dx = k \int f(x) \, dx$, where $k$ is a constant.
Power Rule for Integration: This rule states that $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ for any real number $n \neq -1$. It is used to integrate functions of the form $x^n$.
Simplification: After integrating, it may be necessary to simplify the expression. This involves combining like terms, factoring out constants, and canceling common factors.
By applying these concepts, one can find the antiderivative of a given function step by step, ensuring that the solution is presented in its simplest form.
