Find the Antiderivative f(x)=3x^(1/4)-7x^(3/4)

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 $x^{\frac{1}{4}}$: $3(\frac{4}{5}{x}^{\frac{5}{4}}+C)-\int 7x^{\frac{3}{4}}dx$.

Step 6

Factor out the constant -7 from the second integral: $3(\frac{4}{5}{x}^{\frac{5}{4}}+C)-7\int x^{\frac{3}{4}}dx$.

Step 7

Apply the Power Rule for integration to $x^{\frac{3}{4}}$: $3(\frac{4}{5}{x}^{\frac{5}{4}}+C)-7(\frac{4}{7}{x}^{\frac{7}{4}}+C)$.

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:

1. 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.

2. 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.

3. 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 $n\ne -1$.

4. 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.