Find the Antiderivative f(x)=12x^(5/7)+5x^(-6/7)

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.
$F(x) = \int (12x^{\frac{5}{7}} + 5x^{-\frac{6}{7}}) \, dx$

Step 3:

Decompose the integral into two separate integrals.
$\int 12x^{\frac{5}{7}} \, dx + \int 5x^{-\frac{6}{7}} \, dx$

Step 4:

Extract the constant factor 12 from the first integral.
$12 \int x^{\frac{5}{7}} \, dx + \int 5x^{-\frac{6}{7}} \, dx$

Step 5:

Apply the Power Rule to integrate $x^{\frac{5}{7}}$.
$12 \left( \frac{7}{12} x^{\frac{12}{7}} + C \right) + \int 5x^{-\frac{6}{7}} \, dx$

Step 6:

Extract the constant factor 5 from the second integral.
$12 \left( \frac{7}{12} x^{\frac{12}{7}} + C \right) + 5 \int x^{-\frac{6}{7}} \, dx$

Step 7:

Apply the Power Rule to integrate $x^{-\frac{6}{7}}$.
$12 \left( \frac{7}{12} x^{\frac{12}{7}} + C \right) + 5 \left( 7x^{\frac{1}{7}} + C \right)$

Step 8:

Simplify the expression.

Step 8.1:

Combine the constants and the variable terms.
$12 \left( \frac{7}{12} \right) x^{\frac{12}{7}} + 5 \cdot 7 x^{\frac{1}{7}} + C$

Step 8.2:

Perform the arithmetic operations.

Step 8.2.1:

Multiply 12 by $\frac{7}{12}$.
$\frac{12 \cdot 7}{12} x^{\frac{12}{7}} + 5 \cdot 7 x^{\frac{1}{7}} + C$

Step 8.2.2:

Simplify the fraction.
$7 x^{\frac{12}{7}} + 5 \cdot 7 x^{\frac{1}{7}} + C$

Step 8.2.3:

Multiply 5 by 7.
$7 x^{\frac{12}{7}} + 35 x^{\frac{1}{7}} + C$

Step 9:

Conclude with the antiderivative of $f(x) = 12x^{\frac{5}{7}} + 5x^{-\frac{6}{7}}$.
$F(x) = 7 x^{\frac{12}{7}} + 35 x^{\frac{1}{7}} + C$

Knowledge Notes:

To solve for the antiderivative of a function, we use the process of integration. The antiderivative, also known as the indefinite integral, is a function $F(x)$ such that $F'(x) = f(x)$. The process of finding $F(x)$ involves the following knowledge points:

1. Indefinite Integral: The general form of an indefinite integral is $\int f(x) \, dx$, which represents the family of all antiderivatives of $f(x)$.

2. Power Rule for Integration: When integrating a power of $x$, $x^n$, where $n \neq -1$, the antiderivative is $\frac{x^{n+1}}{n+1} + C$, where $C$ is the constant of integration.

3. Linearity of Integration: The integral of a sum is the sum of the integrals. That is, $\int (af(x) + bg(x)) \, dx = a\int f(x) \, dx + b\int g(x) \, dx$, where $a$ and $b$ are constants.

4. Constants in Integration: Constants can be factored out of the integral. For example, $\int a f(x) \, dx = a \int f(x) \, dx$.

By applying these principles, we can integrate the given function term by term, simplify the resulting expression, and arrive at the antiderivative.