$$ \int20t^2\sqrt[3]{t^{\frac13}}\differentialD t $$

We are tasked with evaluating the integral:

\[
\int 20t^2 \sqrt[3]{t^{\frac{1}{3}}} \, dt
\]

### Step 1: Simplify the integrand
First, let's simplify $ \sqrt[3]{t^{\frac{1}{3}}} $. We know that $ \sqrt[3]{x} = x^{1/3} $, so:

\[
\sqrt[3]{t^{\frac{1}{3}}} = t^{\frac{1}{9}}
\]

Thus, the integrand becomes:

\[
20t^2 \cdot t^{\frac{1}{9}} = 20t^{2 + \frac{1}{9}} = 20t^{\frac{19}{9}}
\]

### Step 2: Apply the power rule for integration
The power rule states that:

\[
\int t^n \, dt = \frac{t^{n+1}}{n+1} + C
\]

In our case, $ n = \frac{19}{9} $. Applying the power rule:

\[
\int 20t^{\frac{19}{9}} \, dt = 20 \cdot \frac{t^{\frac{19}{9} + 1}}{\frac{19}{9} + 1} + C
\]

### Step 3: Simplify the exponent and the fraction
\[
\frac{19}{9} + 1 = \frac{19}{9} + \frac{9}{9} = \frac{28}{9}
\]

Thus, the integral becomes:

\[
20 \cdot \frac{t^{\frac{28}{9}}}{\frac{28}{9}} + C
\]

\[
= 20 \cdot \frac{9}{28} \cdot t^{\frac{28}{9}} + C
\]

\[
= \frac{180}{28} t^{\frac{28}{9}} + C
\]

### Step 4: Simplify the constant
\[
\frac{180}{28} = \frac{90}{14} = \frac{45}{7}
\]

Thus, the final answer is:

\[
\frac{45}{7} t^{\frac{28}{9}} + C
\]

### Final Answer:
\[
\int 20t^2 \sqrt[3]{t^{\frac{1}{3}}} \, dt = \frac{45}{7} t^{\frac{28}{9}} + C
\]

---

Do you want any further details or have any questions?

Here are 5 related questions for further exploration:
1. How would you evaluate $ \int t^2 \sqrt[4]{t^{\frac{1}{2}}} \, dt $?
2. Can you generalize this method to $ \int t^n \sqrt[k]{t^m} \, dt $?
3. What happens if you differentiate the final result to verify the solution?
4. How would you solve $ \int t^{\frac{5}{3}} \, dt $?
5. How does this integral change if the power of $ t $ in the integrand is negative?

**Tip:** When integrating functions with fractional exponents, always simplify the terms before applying the power rule.

Explore how to compute the integral of 20t²√[3]{t^(1/3)} step-by-step by applying the power rule for integration and simplifying fractional exponents. This calculus problem is ideal for undergraduate students looking to master integration techniques.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation