Find the Antiderivative f(x)=3x^(1/4)-5x^(3/4)
Brief Explanation:
The problem is a calculus question that asks for the antiderivative (also known as the indefinite integral) of a given function. Specifically, the function provided is f(x) = 3x^(1/4) - 5x^(3/4), where x^(1/4) represents the fourth root of x, and x^(3/4) represents the fourth root of x cubed. To find the antiderivative, you would apply the rules of integration to each term in the function independently, summing the results to get the general form of the antiderivative. This process involves reversing the differentiation rules, adding a constant of integration at the end since the antiderivative is not unique.
$f \left(\right. x \left.\right) = 3 x^{\frac{1}{4}} - 5 x^{\frac{3}{4}}$
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.
$$F(x) = \int (3x^{\frac{1}{4}} - 5x^{\frac{3}{4}}) \, dx$$
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 $x^{\frac{1}{4}}$.
$$3\left(\frac{4}{5}x^{\frac{5}{4}} + C\right) - \int 5x^{\frac{3}{4}} \, dx$$
Step 6:
Extract the constant $-5$ from the second integral.
$$3\left(\frac{4}{5}x^{\frac{5}{4}} + C\right) - 5\int x^{\frac{3}{4}} \, dx$$
Step 7:
Apply the Power Rule to integrate $x^{\frac{3}{4}}$.
$$3\left(\frac{4}{5}x^{\frac{5}{4}} + C\right) - 5\left(\frac{4}{7}x^{\frac{7}{4}} + C\right)$$
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:
Indefinite Integral: The antiderivative of a function $f(x)$ is represented by $\int f(x) \, dx$. It is the reverse process of differentiation.
Power Rule for Integration: For any real number $n \neq -1$, the integral of $x^n$ with respect to $x$ is $\frac{x^{n+1}}{n+1} + C$, where $C$ is the constant of integration.
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.
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.
Simplification: After applying the Power Rule, we simplify the expression by multiplying the constants and combining like terms.
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.
