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Chapter 4: Problem 39

Determine these indefinite integrals. $$\int\left(x^{2}-\frac{3}{2} \sqrt{x}+x^{-4 / 3}\right) d x$$

Short Answer

Expert verified
\textstyle \frac{x^{3}}{3} - x^{3/2} - 3x^{-1/3} + C\.

Step by step solution

01

- Break Down the Integral

The given integral is \(\textstyle \text{鈭珆(x^{2} - \frac{3}{2} \text{鈭殅{x} + x^{-4/3}) \text{d} x\). Break it down into separate integrals for each term: \(\textstyle \text{鈭珆x^{2} \text{d}x - \frac{3}{2} \text{鈭珆\text{鈭殅{x} \text{d}x + \text{鈭珆x^{-4/3} \text{d}x\).
02

- Evaluate the First Integral

Integrate \(\textstyle x^{2} \) using the power rule \(\textstyle \text{鈭珆x^{n} \text{d}x = \frac{x^{n+1}}{n+1} + C\)\. Applying this, we get: \(\textstyle \text{鈭珆x^{2} \text{d}x = \frac{x^{3}}{3}\).
03

- Evaluate the Second Integral

Rewrite \(\textstyle \text{鈭殅{x} \) as \(\textstyle x^{1/2} \) and use the power rule: \(\textstyle - \frac{3}{2} \text{鈭珆x^{1/2} \text{d}x = - \frac{3}{2} \frac{x^{(1/2)+1}}{(1/2)+1} = - \frac{3}{2} \frac{x^{3/2}}{3/2} = -x^{3/2}\).
04

- Evaluate the Third Integral

For \(\textstyle x^{-4/3}\), apply the power rule: \(\textstyle \text{鈭珆x^{-4/3} \text{d}x = \frac{x^{-4/3+1}}{-4/3+1} = \frac{x^{-1/3}}{-1/3} = -3x^{-1/3}\).
05

- Combine the Results

Sum the results from each of the integrals: \(\textstyle \frac{x^{3}}{3} - x^{3/2} - 3x^{-1/3} + C\), where \(\textstyle C \) is the constant of integration.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integration Techniques
Integration is a crucial skill in calculus, allowing you to find the area under a curve or solve differential equations. One useful method is to break a complex integral into simpler parts. This helps you focus on integrating each term separately.
For example, in the integral \(\textstyle \int(x^{2}-\frac{3}{2} \sqrt{x}+x^{-4/3}) \, dx\), you can break it down like this:
  • \(\int x^{2} \, dx\)
  • -\(\frac{3}{2} \int x^{1/2} \, dx\)
  • \(\int x^{-4/3} \, dx\)
Each of these separate integrals can now be handled using simpler integration techniques.
Power Rule
The power rule is one of the most straightforward techniques to integrate polynomials. It states: \(\textstyle \int x^{n} \, dx = \frac{x^{n+1}}{n+1} + C\).\br>For each term of our original integral:
  • \(\int x^{2} \, dx\) becomes \(\textstyle \frac{x^{3}}{3}\)
  • -\(\frac{3}{2} \int x^{1/2} \, dx\) translates to \(\textstyle - x^{3/2}\)
  • \(\int x^{-4/3} \, dx\) becomes \(\textstyle - 3x^{-1/3}\)
Always remember to increment the exponent by 1 and divide by the new exponent.
Constant of Integration
When integrating, always add a constant of integration, represented as \( C \). This is essential because integration is the inverse operation of differentiation. Constants disappear when differentiating, so they reappear when integrating. In our example, the final result is:
  • \(\frac{x^{3}}{3} - x^{3/2} - 3x^{-1/3} + C\)
The \( + C \) represents any constant value that could have been differentiated away in the original function. Always include this in indefinite integrals.
Integrating Algebraic Expressions
Algebraic expressions often require rewriting or simplification before integrating. For instance, \(\sqrt{x}\) can be rewritten as \(x^{1/2}\) to use the power rule easily.
Here's how the different terms from our example are integrated:
  • \(x^{2} \) simply follows the power rule.
  • \(\text{鈭殅{x} = x^{1/2}, \) making it straightforward to apply the power rule.
  • \(x^{-4/3}\) utilizes the same power rule after simplifying the exponent.
These steps convert a seemingly complex integral into manageable parts. Practice breaking down and rewriting algebraic expressions to make integration easier.

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