Find the Antiderivative f(x)=8x^(3/5)+5x^(-4/5)

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.

$$F(x)=\int (8x^{\frac{3}{5}}+5x^{-\frac{4}{5}})dx$$

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 $x^{\frac{3}{5}}$.

$$8(\frac{5}{8}{x}^{\frac{8}{5}}+C)+\int 5x^{-\frac{4}{5}}dx$$

Step 6

Extract the constant $5$ from the second integral.

$$8(\frac{5}{8}{x}^{\frac{8}{5}}+C)+5\int x^{-\frac{4}{5}}dx$$

Step 7

Apply the Power Rule to integrate $x^{-\frac{4}{5}}$.

$$8(\frac{5}{8}{x}^{\frac{8}{5}}+C)+5(5x^{\frac{1}{5}}+C)$$

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 $F^{\prime }(x)=f(x)$. The general steps for finding the antiderivative include:

1. Setting up the Integral: Write the integral of the function that needs to be solved.

2. Decomposing the Integral: If the function is a sum or difference of functions, separate it into individual integrals.

3. Extracting Constants: Constants can be taken out of the integral, as they do not depend on the variable of integration.

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

5. Simplifying the Expression: After integrating, simplify the expression by combining like terms and canceling common factors.

6. Writing the Final Answer: The final step is to write down the simplified antiderivative, which includes the constant of integration $C$.