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Chapter 8: Problem 9

In Exercises evaluate the integral. $$ \int_{0}^{1} \frac{1}{\left(3 x^{2}+2\right)^{5 / 2}} d x $$

Short Answer

Expert verified
Use substitution and simplify, resulting in an integral in terms of \( u \).

Step by step solution

01

Identify the Type of Integral

Recognize that the integral involves a rational function of the form \( \frac{1}{(u(x))^{n}} \). Here, \( u(x) = 3x^2 + 2 \) and the integral can be simplified using a substitution method.
02

Choose an Appropriate Substitution

Let \( u = 3x^2 + 2 \). Then, \( \frac{du}{dx} = 6x \), giving \( du = 6x \, dx \). Solve for \( dx \):\[ dx = \frac{du}{6x} \].
03

Adjust the Limits of Integration

Change the limits of integration from \( x \) to \( u \). When \( x = 0 \), \( u = 3(0)^2 + 2 = 2 \). When \( x = 1 \), \( u = 3(1)^2 + 2 = 5 \). Thus, the new limits for \( u \) are from 2 to 5.
04

Substitute and Simplify the Integral

Substitute back into the integral with the new variable:\[ \int_{2}^{5} \frac{1}{u^{5/2}} \cdot \frac{1}{6x} \, du. \]Since \( x = \sqrt{\frac{u-2}{3}} \), the integral becomes:\[ \int_{2}^{5} \frac{1}{6\sqrt{\frac{u-2}{3}}} \cdot \frac{1}{u^{5/2}} \, du. \]

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Key Concepts

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

Substitution Method
The substitution method is a powerful tool for solving integrals that might initially seem complex. It involves changing the variable of integration to simplify the problem.
  • Identify a part of the integral that, when changed, makes it easier to integrate. This part is typically a term inside a parentheses or under a radical.
  • In the given exercise, we used the substitution method for the expression \(3x^2 + 2\), setting \(u = 3x^2 + 2\).
  • Calculate the derivative of this substitution term, which is \(\frac{du}{dx} = 6x\), and solve for \(dx\) to express it in terms of \(du\): \(dx = \frac{du}{6x}\).
Using substitution simplifies the integral by removing complicated expressions and reduces the problem to a more manageable equation.
Limits of Integration
When applying the substitution method, it's crucial to adjust the limits of integration to match the new variable.
  • Original limits correspond to the original variable \(x\). For our exercise, these were from 0 to 1.
  • After substitution, recalculate these limits based on the new variable \(u\). For \(x = 0\), \(u\) becomes 2, and for \(x = 1\), \(u\) becomes 5.
  • This results in new limits of integration: from 2 to 5.
Properly changing these limits ensures the integrity of the substituted integral, allowing for a correct evaluation.
Rational Functions
Rational functions are quotients of polynomials and often show up in calculus problems, including integration.
  • These functions can sometimes be difficult to integrate directly due to their complexity. Fortunately, they can often be simplified with techniques like substitution.
  • In this problem, we started with a rational function \( \frac{1}{(3x^2 + 2)^{5/2}} \), which was simplified to \( \frac{1}{u^{5/2}} \) after substitution.
  • Rational functions often allow for flexibility in approach, with methods like partial fraction decomposition sometimes being useful, depending on the problem's form.
Such approaches transform the integral into a simpler form, making evaluation more straightforward and less error-prone.

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