Find the Antiderivative f(x)=8x^(3/5)+5x^(-4/5)
The problem is asking for the calculation of an antiderivative (also known as the indefinite integral) of a given algebraic function. Specifically, the function presented is f(x) = 8x^(3/5) + 5x^(-4/5), which is a sum of two terms where each term is a power function of x. The exponents are given as fractional powers, with one being positive (3/5) and the other negative (-4/5). The question requires finding a new function F(x) such that the derivative of F(x) with respect to x yields the original function f(x). This involves applying the fundamental rules for integration to each term independently and then combining the results to obtain the complete antiderivative.
$f \left(\right. x \left.\right) = 8 x^{\frac{3}{5}} + 5 x^{- \frac{4}{5}}$
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.
$$F(x) = \int (8x^{\frac{3}{5}} + 5x^{-\frac{4}{5}}) \, dx$$
Step 3
Decompose the integral into two separate integrals.
$$\int 8x^{\frac{3}{5}} \, dx + \int 5x^{-\frac{4}{5}} \, dx$$
Step 4
Extract the constant $8$ from the first integral.
$$8\int x^{\frac{3}{5}} \, dx + \int 5x^{-\frac{4}{5}} \, dx$$
Step 5
Apply the Power Rule to integrate $x^{\frac{3}{5}}$.
$$8\left(\frac{5}{8}x^{\frac{8}{5}} + C\right) + \int 5x^{-\frac{4}{5}} \, dx$$
Step 6
Extract the constant $5$ from the second integral.
$$8\left(\frac{5}{8}x^{\frac{8}{5}} + C\right) + 5\int x^{-\frac{4}{5}} \, dx$$
Step 7
Apply the Power Rule to integrate $x^{-\frac{4}{5}}$.
$$8\left(\frac{5}{8}x^{\frac{8}{5}} + C\right) + 5\left(5x^{\frac{1}{5}} + C\right)$$
Step 8
Simplify the expression.
Step 8.1
Multiply the constants inside the parentheses.
$$8\left(\frac{5}{8}\right)x^{\frac{8}{5}} + 5 \cdot 5x^{\frac{1}{5}} + C$$
Step 8.2
Simplify further.
Step 8.2.1
Combine $8$ and $\frac{5}{8}$.
$$\frac{8 \cdot 5}{8}x^{\frac{8}{5}} + 5 \cdot 5x^{\frac{1}{5}} + C$$
Step 8.2.2
Simplify the fraction.
$$\frac{40}{8}x^{\frac{8}{5}} + 5 \cdot 5x^{\frac{1}{5}} + C$$
Step 8.2.3
Cancel out common factors.
Step 8.2.3.1
Factor out $8$ from $40$.
$$\frac{8 \cdot 5}{8}x^{\frac{8}{5}} + 5 \cdot 5x^{\frac{1}{5}} + C$$
Step 8.2.3.2
Eliminate the common factors.
Step 8.2.3.2.1
Factor out $8$ from $8$.
$$\frac{8 \cdot 5}{8(1)}x^{\frac{8}{5}} + 5 \cdot 5x^{\frac{1}{5}} + C$$
Step 8.2.3.2.2
Cancel the common factor.
$$\frac{\cancel{8} \cdot 5}{\cancel{8} \cdot 1}x^{\frac{8}{5}} + 5 \cdot 5x^{\frac{1}{5}} + C$$
Step 8.2.3.2.3
Rewrite the simplified expression.
$$\frac{5}{1}x^{\frac{8}{5}} + 5 \cdot 5x^{\frac{1}{5}} + C$$
Step 8.2.3.2.4
Divide $5$ by $1$.
$$5x^{\frac{8}{5}} + 5 \cdot 5x^{\frac{1}{5}} + C$$
Step 8.2.4
Multiply $5$ by $5$.
$$5x^{\frac{8}{5}} + 25x^{\frac{1}{5}} + C$$
Step 9
Conclude with the antiderivative of $f(x) = 8x^{\frac{3}{5}} + 5x^{-\frac{4}{5}}$.
$$F(x) = 5x^{\frac{8}{5}} + 25x^{\frac{1}{5}} + C$$
Knowledge Notes:
The process of finding the antiderivative, also known as the indefinite integral, involves reversing the differentiation process. The antiderivative of a function $f(x)$ is another function $F(x)$ such that $F'(x) = f(x)$. The general steps for finding the antiderivative include:
Setting up the Integral: Write the integral of the function that needs to be solved.
Decomposing the Integral: If the function is a sum or difference of functions, separate it into individual integrals.
Extracting Constants: Constants can be taken out of the integral, as they do not depend on the variable of integration.
Applying the Power Rule: The Power Rule for integration states that $\int x^n \, dx = \frac{1}{n+1}x^{n+1} + C$, where $n \neq -1$ and $C$ is the constant of integration.
Simplifying the Expression: After integrating, simplify the expression by combining like terms and canceling common factors.
Writing the Final Answer: The final step is to write down the simplified antiderivative, which includes the constant of integration $C$.
