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Magazine Chapter 8: Problem 40
Use a substitution to change the integral into one you can find in the table. Then evaluate the integral. $$\int x^{2} \sqrt{2 x-x^{2}} d x$$
Short Answer
Expert verified
Use substitution \( u = 2x - x^2 \) to find an integral that matches a standard form, then evaluate.
Step by step solution
01
Choose a Substitution
Let's select a substitution to simplify the integral. The integrand involves the expression \( \sqrt{2x - x^2} \), which suggests a trigonometric substitution or completing the square. Instead, let's use the substitution \( u = 2x - x^2 \), which implies \( du = (2 - 2x)dx = 2(1-x)dx \). Solving for \( dx \), we get \( dx = \frac{du}{2(1-x)} \). We will express \( x \) in terms of \( u \) next.
02
Simplify Substitution
From the substitution \( u = 2x - x^2 \), we note that \( u = -(x^2 - 2x) = -(x-1)^2 + 1 \), and hence \( x = 1 \pm \sqrt{1-u} \). We will use the positive root \( x = 1 + \sqrt{1-u} \). Now express \( dx \) in terms of \( du \): since \( du = (2 - 2x)dx \), \( dx = \frac{du}{2 - 2x} = \frac{du}{2(1-x)} \). Substitute \( x = 1 + \sqrt{1-u} \) back into the expression for \( du \) to simplify further.
03
Substitute and Simplify the Integral
Substituting \( u = 2x - x^2 \) and \( dx = \frac{du}{2(1-x)} \) back into the integral gives us a new integral with respect to \( u \). Substitute \( x = 1+\sqrt{1-u} \), and modify the numerator \( x^2 = (1+\sqrt{1-u})^2 \), leading to a complex expression. The key is to simplify this into a usable form with our substitution.
04
Solve the Integral
After simplification, integrate with respect to \( u \). Find the resulting integral in a standard integral table if it matches a known form.
05
Back Substitute
Finally, substitute back the variable \( x \) in terms of \( u \) to provide the solution in the original variable \( x \). Ensure any constants of integration are clearly noted.
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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 fundamental technique in calculus used to simplify complex integrals by introducing a new variable. This method often turns a difficult integral into one that is more manageable or even straightforward.
- Choose a substitution that simplifies the integral. For instance, if your integrand contains a square root or a polynomial that seems unwieldy, consider letting a part of the integrand be your new variable.
- In this problem, the integrand \(\sqrt{2x - x^2}\) suggests a substitution of \(u = 2x - x^2\), reducing the complexity of the expression inside the square root.
- After choosing your substitution, express the differential \(dx\) in terms of \(du\), as done here with \(dx = \frac{du}{2(1-x)}\).
Integral Tables
Integral tables are valuable resources in calculus that provide formulae for a wide array of standard integrals. They allow students to find integrals quickly without performing step-by-step integration every time.
- Once a substitution is made, the resulting integral can often match a formula from these tables.
- In this exercise, after simplifying the integral using substitution, you can look in the tables for a matching formula.
- If the direct form is not available, slight algebraic manipulations may adjust your integral to fit a known integral form from the table.
Trigonometric Substitution
Trigonometric substitution is a technique used when dealing with integrals involving square roots, especially those that resemble the forms of \(\sqrt{a^2 - x^2}\), \(\sqrt{x^2 - a^2}\), or \(\sqrt{x^2 + a^2}\).
- This method involves substituting trigonometric identities for polynomial expressions to simplify integration.
- In the given exercise, while not ultimately used, this substitution was considered due to the square root present in the integrand.
- A trigonometric substitution could help express roots and other complex parts of the integrand in terms of sines or cosines, making the integral approachable.
Completing the Square
Completing the square is a useful algebraic technique for rewriting quadratic expressions into a perfect square trinomial. This technique is often used in integration to simplify expressions under square roots.
- For instance, given an expression like \(2x - x^2\), completing the square involves rewriting it in the form \(-(x-1)^2 + 1\).
- This transformation clarifies the behavior of the expression and can reveal simpler substitution options.
- In the exercise, completing the square helps in expressing the integrand in a form where substitution becomes clearer and more straightforward.
