Find the Antiderivative f(z) = fourth root of z^3

Solution:

Step 1:

Identify the antiderivative $F(z)$ by integrating the function $f(z)$.

$$ F(z) = \int f(z) \, dz $$

Step 2:

Write down the integral that needs to be solved.

$$ F(z) = \int \sqrt[4]{z^3} \, dz $$

Step 3:

Convert the fourth root into an exponent.

$$ \int z^{\frac{3}{4}} \, dz $$

Step 4:

Apply the Power Rule for integration to find the antiderivative.

$$ \frac{4}{7} z^{\frac{7}{4}} + C $$

Step 5:

Present the final form of the antiderivative of $f(z) = \sqrt[4]{z^3}$.

$$ F(z) = \frac{4}{7} z^{\frac{7}{4}} + C $$

Knowledge Notes:

1. Indefinite Integral: The process of finding the antiderivative of a function is known as finding the indefinite integral. The antiderivative is a function whose derivative is the original function.

2. Integral Setup: To solve an integral, we first express the function in a form that is suitable for integration, often involving algebraic manipulation.

3. Exponent Conversion: Roots can be converted to fractional exponents, which is a useful step for integration. The nth root of $a^x$ is written as $a^{x/n}$.

4. Power Rule for Integration: The Power Rule states that the integral of $z^n$ with respect to $z$ is $\frac{1}{n+1} z^{n+1}$, provided $n \neq -1$. This is a fundamental rule in calculus used for finding antiderivatives.

5. Integration Constant: When finding the indefinite integral, we add a constant $C$ because the derivative of a constant is zero, and thus the antiderivative is not unique without this constant.

6. Latex Formatting: In the solution, Latex is used to render mathematical expressions and symbols for clarity and precision.