Find the Antiderivative f(x)=(x/12)^15

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.

$$ F(x) = \int \left(\frac{x}{12}\right)^{15} dx $$

Step 3:

Use substitution: Let $u = \frac{x}{12}$. Then, calculate $du$ in terms of $dx$.

Step 3.1:

Define the substitution $u = \frac{x}{12}$ and find $\frac{du}{dx}$.

Step 3.1.1:

Take the derivative of $\frac{x}{12}$ with respect to $x$.

$$ \frac{d}{dx} \left(\frac{x}{12}\right) $$

Step 3.1.2:

Since $\frac{1}{12}$ is a constant, its derivative with respect to $x$ is $\frac{1}{12} \frac{d}{dx}(x)$.

$$ \frac{1}{12} \frac{d}{dx}(x) $$

Step 3.1.3:

Apply the Power Rule, which states that $\frac{d}{dx}(x^n) = n x^{n-1}$, where $n = 1$.

$$ \frac{1}{12} \cdot 1 $$

Step 3.1.4:

Multiply $\frac{1}{12}$ by $1$ to get the derivative of $u$.

$$ \frac{1}{12} $$

Step 3.2:

Substitute $u$ and $du$ into the integral.

$$ \int u^{15} \cdot 12 \, du $$

Step 4:

Simplify the integral.

Step 4.1:

Multiply $u^{15}$ by the reciprocal of $\frac{1}{12}$.

$$ \int u^{15} \cdot (1 \cdot 12) \, du $$

Step 4.2:

Simplify the multiplication.

$$ \int u^{15} \cdot 12 \, du $$

Step 4.3:

Rearrange the constant and the variable.

$$ \int 12 u^{15} \, du $$

Step 5:

Extract the constant $12$ from the integral.

$$ 12 \int u^{15} \, du $$

Step 6:

Integrate $u^{15}$ using the Power Rule.

$$ 12 \left(\frac{1}{16} u^{16} + C\right) $$

Step 7:

Simplify the expression.

Step 7.1:

Rewrite the expression.

$$ 12 \left(\frac{1}{16}\right) u^{16} + C $$

Step 7.2:

Combine constants.

Step 7.2.1:

Reduce the fraction $\frac{12}{16}$.

$$ \frac{12}{16} u^{16} + C $$

Step 7.2.2:

Simplify by canceling common factors.

Step 7.2.2.1:

Factor out a $4$ from the numerator.

$$ \frac{4(3)}{16} u^{16} + C $$

Step 7.2.2.2:

Cancel the $4$ in the numerator and denominator.

$$ \frac{4 \cdot 3}{4 \cdot 4} u^{16} + C $$

Step 7.2.2.2.1:

Eliminate the common factor.

$$ \frac{\cancel{4} \cdot 3}{\cancel{4} \cdot 4} u^{16} + C $$

Step 7.2.2.2.2:

Write the simplified expression.

$$ \frac{3}{4} u^{16} + C $$

Step 8:

Substitute back the original variable $x$ for $u$.

$$ \frac{3}{4} \left(\frac{x}{12}\right)^{16} + C $$

Step 9:

Reorganize the terms for clarity.

$$ \frac{3}{4} \left(\frac{1}{12} x\right)^{16} + C $$

Step 10:

Present the final antiderivative of the function $f(x) = \left(\frac{x}{12}\right)^{15}$.

$$ F(x) = \frac{3}{4} \left(\frac{1}{12} x\right)^{16} + C $$

Knowledge Notes:

The process of finding an antiderivative involves integrating a given function. The antiderivative, also known as the indefinite integral, of a function $f(x)$ is denoted by $F(x)$ and is defined such that $F'(x) = f(x)$.

The Power Rule for integration states that the integral of $x^n$ with respect to $x$ is $\frac{1}{n+1}x^{n+1}$, provided that $n \neq -1$.

Substitution is a method used in integration to simplify the integral by changing variables. It often involves letting $u$ be a function of $x$ and then expressing $dx$ in terms of $du$.

Constants can be factored out of integrals, and the integral of a constant times a function is equal to the constant times the integral of the function.

When simplifying fractions, common factors in the numerator and denominator can be canceled out to reduce the fraction to its simplest form.

After integrating with a substitution, it is important to substitute back to the original variable to express the antiderivative in terms of the original variable given in the problem.