Find the Antiderivative f(x)=3x^(1/4)-5x^(3/4)

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.

$$F(x)=\int (3x^{\frac{1}{4}}-5x^{\frac{3}{4}})dx$$

Step 3:

Decompose the integral into two separate integrals.

$$\int 3x^{\frac{1}{4}}dx-\int 5x^{\frac{3}{4}}dx$$

Step 4:

Extract the constant $3$ from the first integral.

$$3\int x^{\frac{1}{4}}dx-\int 5x^{\frac{3}{4}}dx$$

Step 5:

Apply the Power Rule to integrate $x^{\frac{1}{4}}$.

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

Step 6:

Extract the constant $-5$ from the second integral.

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

Step 7:

Apply the Power Rule to integrate $x^{\frac{3}{4}}$.

$$3(\frac{4}{5}{x}^{\frac{5}{4}}+C)-5(\frac{4}{7}{x}^{\frac{7}{4}}+C)$$

Step 8:

Proceed to simplify the expression.

Step 8.1:

Combine terms.

$$\frac{12}{5}{x}^{\frac{5}{4}}-\frac{20}{7}{x}^{\frac{7}{4}}+C$$

Step 8.2:

Final simplification.

Step 8.2.1:

Multiply $3$ by $\frac{4}{5}$.

$$\frac{12}{5}{x}^{\frac{5}{4}}-\frac{20}{7}{x}^{\frac{7}{4}}+C$$

Step 8.2.2:

Multiply $-5$ by $\frac{4}{7}$.

$$\frac{12}{5}{x}^{\frac{5}{4}}-\frac{20}{7}{x}^{\frac{7}{4}}+C$$

Step 8.2.3:

Present the final antiderivative.

$$F(x)=\frac{12}{5}{x}^{\frac{5}{4}}-\frac{20}{7}{x}^{\frac{7}{4}}+C$$

Step 9:

Conclude with the antiderivative of the function $f(x)=3x^{\frac{1}{4}}-5x^{\frac{3}{4}}$.

$$F(x)=\frac{12}{5}{x}^{\frac{5}{4}}-\frac{20}{7}{x}^{\frac{7}{4}}+C$$

Knowledge Notes:

To solve for the antiderivative (indefinite integral) of a function, we follow these key points:

1. Indefinite Integral: The antiderivative of a function $f(x)$ is represented by $\int f(x)dx$. It is the reverse process of differentiation.

2. Power Rule for Integration: For any real number $n\ne -1$, the integral of $x^n$ with respect to $x$ is $\frac{x^{n+1}}{n+1}+C$, where $C$ is the constant of integration.

3. Linearity of the Integral: The integral of a sum or difference of functions is the sum or difference of their integrals. This property allows us to split the integral of $3x^{\frac{1}{4}}-5x^{\frac{3}{4}}$ into two separate integrals.

4. Constant Multiple Rule: A constant multiplied by a function can be taken outside the integral. This is why we can move the constants $3$ and $-5$ outside their respective integrals.

5. Simplification: After applying the Power Rule, we simplify the expression by multiplying the constants and combining like terms.

6. Constant of Integration: Since the antiderivative is not unique (due to the derivative of a constant being zero), we add a constant $C$ to the result to represent the family of all antiderivatives of the function.