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Magazine Chapter 8: Problem 10
In Exercises evaluate the integral. $$ \int \frac{x^{2}}{\left(x^{2}-2\right)^{5 / 2}} d x $$
Short Answer
Expert verified
The integral evaluates to \(-((x^{2}-2)^{-1/2} + 2(x^{2}-2)^{-3/2}) + C\).
Step by step solution
01
Recognize the integral form
The given integral is \( \int \frac{x^{2}}{(x^{2}-2)^{5/2}} \, dx \). It involves a rational function with a square root in the denominator.
02
Choose a substitution
Since the integrand involves \((x^{2} - 2)^{5/2}\), a good substitution is \( u = x^{2} - 2 \). This simplifies the integrand, allowing easier integration.
03
Compute the derivative of substitution
Differentiate the substitution \( u = x^{2} - 2 \) to find \( du = 2x \, dx \). Hence, we can write \( x \, dx = \frac{1}{2} du \).
04
Adjust the integral for substitution
Rewrite the integral in terms of \( u \) and \( du \). The expression becomes \( \int \frac{x}{u^{5/2}} \times x \, dx = \int \frac{x^2}{u^{5/2}} \, dx = \int \frac{u + 2}{u^{5/2}} \, \frac{1}{2} du \).
05
Split the integral
Separate the integral into two simpler integrals: \( \frac{1}{2} \left( \int \frac{1}{u^{3/2}} \, du + \int \frac{2}{u^{5/2}} \, du \right) \).
06
Integrate each term separately
Compute each separate integral:1. \( \int u^{-3/2} \, du = -2u^{-1/2} \)2. \( \int 2u^{-5/2} \, du = -4u^{-3/2} \)
07
Simplify and combine results
Substitute current values back into place to derive the final combined integral expression: \( \frac{1}{2} \left( -2u^{-1/2} - 4u^{-3/2} \right) = -u^{-1/2} - 2u^{-3/2} \).
08
Substitute back for original variable
Replace \( u \) with \( x^{2} - 2 \) to return to original variable:\( -((x^{2}-2)^{-1/2} + 2(x^{2}-2)^{-3/2}) \ + C \), where \( C \) is the integration constant.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Substitution Method
In calculus, the substitution method is a powerful technique used to simplify integrals by making a change of variables. This approach can transform a complicated integral into a much simpler one.
- First, identify a part of the integrand that can be replaced with a new variable. This substitution should simplify the integral.
- Next, replace this part of the integrand with the new variable, say \( u \). Then, differentiate \( u \) to find \( du \), which helps in expressing the entire integral in terms of \( u \).
- The goal is to simplify the integral so it becomes easier to solve, often turning it into a basic integral form you recognize.
Definite Integrals
Definite integrals are used to calculate the actual value representing the area under a curve between two specific limits. They contrast with indefinite integrals, which represent a general form of a function's antiderivative.
- When solving definite integrals, you can use the substitution method to simplify the integral before applying the limits of integration.
- Once you integrate, don't forget to evaluate the integral at the upper and lower limits and subtract the latter value from the former.
- Definite integrals provide precision in calculating total changes or areas, offering insights into many physical and geometric problems.
Rational Functions
A rational function is a ratio of two polynomials. Understanding these functions is crucial, as they often appear in calculus problems, particularly in integration.
- These functions can involve square roots and other complexities, like the one in this task with the square root in the denominator.
- When confronted with rational functions in integrals, seek to simplify them, sometimes by using techniques like partial fraction decomposition or substitution.
- Simplifying helps transform them into a form that's easier to integrate, as complex fractions or radical expressions can often be unwieldy.
