Find the Antiderivative f(x)=3x^(1/4)-7x^(3/4)
The problem is asking to determine the antiderivative, also known as the indefinite integral, of the function f(x) which is given by the expression 3x^(1/4) - 7x^(3/4). This involves finding a function F(x) such that its derivative with respect to x is the original function f(x). It is a calculus problem that seeks to reverse the process of differentiation. The antiderivative is typically expressed with a "+ C" at the end of the expression, where C represents a constant of integration.
$f \left(\right. x \left.\right) = 3 x^{\frac{1}{4}} - 7 x^{\frac{3}{4}}$
Answer
Solution:
Step 1
To find the antiderivative of $f(x)$, integrate the function with respect to $x$: $F(x) = \int f(x) \, dx$.
Step 2
Write down the integral that needs to be solved: $F(x) = \int (3x^{\frac{1}{4}} - 7x^{\frac{3}{4}}) \, dx$.
Step 3
Decompose the integral into two separate integrals: $\int 3x^{\frac{1}{4}} \, dx - \int 7x^{\frac{3}{4}} \, dx$.
Step 4
Factor out the constant 3 from the first integral: $3\int x^{\frac{1}{4}} \, dx - \int 7x^{\frac{3}{4}} \, dx$.
Step 5
Apply the Power Rule for integration to $x^{\frac{1}{4}}$: $3\left(\frac{4}{5}x^{\frac{5}{4}} + C\right) - \int 7x^{\frac{3}{4}} \, dx$.
Step 6
Factor out the constant -7 from the second integral: $3\left(\frac{4}{5}x^{\frac{5}{4}} + C\right) - 7\int x^{\frac{3}{4}} \, dx$.
Step 7
Apply the Power Rule for integration to $x^{\frac{3}{4}}$: $3\left(\frac{4}{5}x^{\frac{5}{4}} + C\right) - 7\left(\frac{4}{7}x^{\frac{7}{4}} + C\right)$.
Step 8
Simplify the expression.
Step 8.1
Combine the constants with their respective powers of $x$: $3\left(\frac{4}{5}\right)x^{\frac{5}{4}} - 7\left(\frac{4}{7}\right)x^{\frac{7}{4}} + C$.
Step 8.2
Further simplify the expression.
Step 8.2.1
Multiply 3 by $\frac{4}{5}$: $\frac{12}{5}x^{\frac{5}{4}} - 7\left(\frac{4}{7}\right)x^{\frac{7}{4}} + C$.
Step 8.2.2
Multiply -7 by $\frac{4}{7}$: $\frac{12}{5}x^{\frac{5}{4}} - 4x^{\frac{7}{4}} + C$.
Step 8.2.3
The common factors in the coefficients have already been canceled out.
Step 8.2.4
The final expression is $\frac{12}{5}x^{\frac{5}{4}} - 4x^{\frac{7}{4}} + C$.
Step 9
The antiderivative of $f(x) = 3x^{\frac{1}{4}} - 7x^{\frac{3}{4}}$ is $F(x) = \frac{12}{5}x^{\frac{5}{4}} - 4x^{\frac{7}{4}} + C$.
Knowledge Notes:
Indefinite Integral: The process of finding the antiderivative of a function is called indefinite integration. The antiderivative is a function whose derivative is the original function.
Constant Factor Rule: When integrating, a constant can be factored out of the integral. This is because the derivative of a constant times a function is the constant times the derivative of the function.
Power Rule for Integration: The Power Rule states that the integral of $x^n$ with respect to $x$ is $\frac{1}{n+1}x^{n+1}$, provided that $n \neq -1$.
Integration Constant: When finding the indefinite integral, an arbitrary constant $C$ is added to the result because the derivative of a constant is zero, and thus the original function could have any constant added to it.
