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Chapter 5: Problem 25

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

Short Answer

Expert verified
\( \int (x^2 - \frac{3}{2} \sqrt{x} + x^{-4/3}) \ dx = \frac{x^3}{3} - x^{3/2} - 3 x^{-1/3} + C \)

Step by step solution

01

Understand the Problem

The problem requires evaluating the integral of the function: \ \[ \int (x^2 - \frac{3}{2} \sqrt{x} + x^{- \frac{4}{3}}) \ dx \ \]
02

Simplify the Integrand

Rewrite the integrand in a form that makes it easier to integrate: \ \[ x^2 - \frac{3}{2} x^{1/2} + x^{-4/3} \ \]
03

Integrate Term by Term

Integrate each term separately. Start with the first term: \ \[ \int x^2 \ dx = \frac{x^{3}}{3} \ \ \] For the second term: \ \[ \int - \frac{3}{2} x^{1/2} \ dx = - \frac{3}{2} \int x^{1/2} \ dx = - \frac{3}{2} \cdot \frac{2 x^{3/2}}{3} = -x^{3/2} \ \ \] For the third term: \ \[ \int x^{-4/3} \ dx = \int x^{-4/3} \ dx = \frac{x^{-1/3}}{-1/3} = -3 x^{-1/3} \ \ \]
04

Combine the Results

Now combine the results of each term: \ \[ \frac{x^3}{3} - x^{3/2} - 3 x^{-1/3} + C \ \]

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Key Concepts

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

definite and indefinite integrals
Integrals are essential in calculus for calculating areas, volumes, and more. There are two main types: definite and indefinite integrals. The integral in this exercise is an indefinite integral because it doesn't have specific limits of integration, like two numbers at the bottom and top of the integral sign. Instead, indefinite integrals find the antiderivative of a function, adding a constant of integration, often represented by C. A definite integral, on the other hand, calculates the exact area under a curve between two points and doesn't include the constant C.
integration techniques
When solving integrals, various techniques can be applied based on the form of the function. Some common methods include:
  • Substitution: Used when a change of variable simplifies the integral.
  • Integration by Parts: Useful for products of functions.
  • Partial Fraction Decomposition: Breaks down complex rational functions.
  • Trigonometric Substitution: Applies to integrands involving square roots.

For the given exercise, we use term-by-term integration, allowing us to handle each part of the polynomial separately, simplifying the whole process.
polynomial integration
Polynomial integration is straightforward since polynomials are sums of power functions. The rule for integrating a power function is to add one to the exponent and then divide by the new exponent. For example:
  • The integral of \( x^{n} \) is \( \frac{x^{n+1}}{n+1} \)'
  • The steps are simple, making polynomial integration one of the easiest forms of integration.

In the exercise, you see functions like \( x^2, x^{1/2}, \) and some with negative exponents. Each term is integrated individually using the power rule.
integration of power functions
Integration of power functions follows a general rule:

  • For any power function \( x^{n} \), the integral is \( \frac{x^{n+1}}{n+1} \). This works for both positive and negative exponents.
  • For instance, in the exercise:
  • Integrating \( x^2 \): \( \frac{x^{3}}{3} \)
  • Integrating \( x^{1/2} \): \( \frac{2x^{3/2}}{3} \),
  • We multiply by the term's coefficient (here, -3/2).
  • For \( x^{-4/3} \), the integration is \( \frac{x^{-1/3}}{-1/3} = -3 x^{-1/3} \).

Combining these, we get the final answer \( \frac{x^{3}}{3} - x^{3/2} - 3 x^{-1/3} + C \).

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