Problem 32 - Calculus Early Transcendentals

Question

Determine the following indefinite integrals. Check your work by differentiation. $$\int\left(4 z^{1 / 3}-z^{-1 / 3}\right) d z$$

Step-by-Step Solution

Answer

Question: Find the indefinite integral of the function $$\int\left(4 z^{1 / 3}-z^{-1 / 3}\right) d z$$. Answer: The indefinite integral of the given function is $$3z^{4/3} - \frac{3}{2}z^{2/3} + C$$.

Step 1: Identify the terms to be integrated separately

Given the function $$\int\left(4 z^{1 / 3}-z^{-1 / 3}\right) d z$$, we can separate the two terms and integrate them separately. They are: 1. $$4z^\frac{1}{3}$$ 2. $$-z^{-\frac{1}{3}}$$

Step 2: Integrate the first term

We will integrate $$4z^{1/3}$$ with respect to z. Use the power rule of integration, which states that $$\int x^n dx = \frac{x^{n+1}}{n+1} + C$$, where n is a constant. $$\int 4z^{1/3} dz = 4\int z^{1/3} dz = 4\cdot\frac{z^{1/3+1}}{1/3+1}+C_1$$ Simplify the expression: $$= 4\cdot\frac{z^{4/3}}{4/3}+C_1$$ $$= 3z^{4/3} + C_1$$

Step 3: Integrate the second term

Next, we will integrate $$-z^{-1/3}$$ with respect to z. Again, use the power rule of integration: $$\int -z^{-1/3} dz = -\int z^{-1/3} dz = -\frac{z^{-1/3+1}}{-1/3+1}+C_2$$ Simplify the expression: $$= -\frac{z^{2/3}}{2/3}+C_2$$ $$= -\frac{3}{2}z^{2/3} + C_2$$

Step 4: Combine the results

Now, we will combine the results from Step 2 and Step 3 to find the indefinite integral: $$\int(4 z^{1 / 3}-z^{-1 / 3}) d z = 3z^{4/3} - \frac{3}{2}z^{2/3} + C$$ where $$C = C_1 + C_2$$ is the constant of integration.

Step 5: Verify the answer by differentiation

To check our work, we will differentiate the result and see if it matches the original function. $$\frac{d}{dz} (3z^{4/3} - \frac{3}{2}z^{2/3} + C)$$ Using the power rule for differentiation, we get: $$\frac{d}{dz} (3z^{4/3}) - \frac{d}{dz} (\frac{3}{2}z^{2/3})$$ $$= 4z^{1/3} - z^{-1/3}$$ The differentiated result matches the original function, so our integration is correct.