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Chapter 7: Problem 9

Find the antiderivatives of the functions: \(\frac{2}{x \sqrt{x}}\)

Short Answer

Expert verified
The antiderivative is \(-\frac{4}{\sqrt{x}} + C\).

Step by step solution

01

Rewrite the expression

The given function is \( \frac{2}{x \sqrt{x}} \). Simplify \( \sqrt{x} \) as \( x^{1/2} \) and rewrite the expression as \( \frac{2}{x \cdot x^{1/2}} \). This simplifies to \( \frac{2}{x^{3/2}} \) because \( x \cdot x^{1/2} = x^{1 + 1/2} = x^{3/2} \).
02

Express with negative exponent

Re-express \( \frac{2}{x^{3/2}} \) in terms of a negative exponent: \( 2x^{-3/2} \). This is convenient for integrating polynomial expressions.
03

Apply the power rule for integration

Integrate \( 2x^{-3/2} \) using the power rule \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \). Here, \( n = -3/2 \). Adding 1 to the exponent: \( -3/2 + 1 = -1/2 \).
04

Calculate the antiderivative

Substitute the new exponent into the formula: \( \int 2x^{-3/2} \, dx = \frac{2x^{-1/2}}{-1/2} + C \). Simplifying, \( \frac{2}{-1/2} = -4 \), so the antiderivative is \( -4x^{-1/2} + C \).
05

Rewrite in simplified form

Rewrite \( -4x^{-1/2} + C \) in terms of radicals: \( -\frac{4}{\sqrt{x}} + C \). This is the antiderivative expressed in its simplest form.

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Key Concepts

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

Power Rule for Integration
The power rule for integration is a straightforward technique used to find the antiderivative of polynomial functions. It is similar to the power rule for differentiation but reversed. The general formula for the power rule is:
  • \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \)
Here, \( n \) represents the exponent of \( x \) and \( C \) is the constant of integration. To apply this rule, simply increase the exponent by one and divide by the new exponent.
In this problem, the function \( 2x^{-3/2} \) is integrated. By adding 1 to the exponent \( -3/2 \), you obtain \( -1/2 \). Then, dividing by \( -1/2 \), the antiderivative becomes:
  • \( -4x^{-1/2} + C \)
Applying the power rule can simplify the process of integrating functions with different exponents.
Integrals with Negative Exponents
Integrating a function with a negative exponent is a common task in calculus. Negative exponents indicate reciprocals or divisions by a power of a number. In the given exercise, the function \( \frac{2}{x^{3/2}} \) is re-expressed using a negative exponent:
  • \( 2x^{-3/2} \)
This transformation is crucial as it allows the use of the power rule for integration, which is easier to apply to polynomial forms than fractions or radicals.
When integrating negative exponents, always remember to:
  • Convert the fractional expression to a power with a negative exponent.
  • Use the power rule, keeping in mind that the exponent must not equal \( -1 \), as that would require a different approach (the natural logarithm).
Through these steps, the integration process becomes more manageable and straightforward.
Simplifying Expressions with Radicals
Simplifying expressions with radicals often involves converting radicals to exponents. This is a crucial step in calculus as it helps in both differentiating and integrating functions.
In this exercise, \( \sqrt{x} \) is represented as \( x^{1/2} \). Simplifying the original expression \( \frac{2}{x \sqrt{x}} \) involves the following:
  • Expressing \( \sqrt{x} \) as \( x^{1/2} \).
  • Combining exponents: \( x \cdot x^{1/2} = x^{3/2} \).
  • Rewriting as \( \frac{2}{x^{3/2}} \).
After simplification, you can more easily use integration rules. The expression is then simplified further into terms with negative exponents, making the integration process smoother. By mastering radical simplifications, you pave the way to more advanced calculus techniques with confidence.

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