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Chapter 8: Problem 2

What change of variables is suggested by an integral containing \(\sqrt{x^{2}+36 ?}\)

Short Answer

Expert verified
Question: Determine the appropriate change of variables for an integral containing the square root of a sum with the form \(\sqrt{x^2+36}\). Answer: The appropriate change of variables for an integral containing the square root of a sum with the form \(\sqrt{x^2+36}\) is \(x = 6\tan\theta\), as it simplifies the expression and helps in solving the integral more easily.

Step by step solution

01

Identify the appropriate trigonometric substitution

The expression \(\sqrt{x^2+36}\) can be rewritten as \(\sqrt{x^2+6^2}\), which resembles the Pythagorean identity \(sin^2\theta + cos^2\theta = 1\). By analogy with the Pythagorean theorem, we will use the tangent function as the substitution because of its connection with the opposite and adjacent sides of a right triangle. So we can make the substitution \(x = 6\tan\theta\), where \(x\) is the length of the opposite side and \(6\) is the length of the adjacent side.
02

Differentiate the substitution

To perform the substitution, we also need the differential of x in terms of theta. Differentiate the expression \(x = 6\tan\theta\) with respect to \(\theta\) to find the differential: \(\frac{dx}{d\theta} = 6\sec^2\theta\). Now, we can write the differential as \(dx = 6\sec^2\theta d\theta\).
03

Substitute and simplify

Now that we have our substitution and differential, we can substitute these expressions into the integral (let's say we have an integral of the form \(\int f(x) \sqrt{x^2+36}dx\)): \(\int f(x)\sqrt{x^2+36}dx = \int f(6\tan\theta)\sqrt{(6\tan\theta)^2+36} (6\sec^2\theta d\theta)\). This will simplify the expression inside the square root because we now have: \(\sqrt{(6\tan\theta)^2+36} = \sqrt{36\tan^2\theta+36} = \sqrt{36(\tan^2\theta+1)} = \sqrt{36\sec^2\theta} = 6\sec\theta\). So now, our integral becomes: \(\int f(6\tan\theta) (6\sec\theta) (6\sec^2\theta d\theta)\).
04

Calculate the new integral and find the original solution

Solve the new integral with respect to \(\theta\) by referring to the original exercise if necessary. After that, replace \(\theta\) with the original variable \(x\) by using the inverse of the substitution we made in step 1. In our case, we know that \(x = 6\tan\theta\), so we can find \(\theta\) by rewriting it as \(\theta = \arctan(\frac{x}{6})\). This step will depend on the actual function \(f(x)\) in the integral. In conclusion, when encountering the expression \(\sqrt{x^2+36}\) in an integral, a change of variables using the substitution \(x = 6\tan\theta\) is suggested, and then proceed with solving the integral in terms of \(\theta\). Once that is done, replace \(\theta\) with \(x\) to find the solution to the original integral.

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Key Concepts

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

Integral Calculus
Integral calculus is a branch of mathematics focused on the processes of integration and the finding of a function's antiderivative. In simple terms, it allows us to determine the accumulation of quantities, such as areas under curves, total distance traveled, or the net amount of substance dissolved over time.

When we encounter an integral with a square root like \(\sqrt{x^{2}+36}\), it becomes a challenge because it doesn't match standard integral forms. In this case, we use techniques such as trigonometric substitution to rewrite the integral into a more manageable form.
Pythagorean Identity
The Pythagorean identity is a fundamental relationship in trigonometry expressed as \(\sin^{2}\theta + \cos^{2}\theta = 1\). It's derived from the Pythagorean theorem, which relates the lengths of the sides of a right-angled triangle.

In the context of trigonometric substitution for integration, the Pythagorean identity helps us recognize patterns that resemble the formula of a right triangle's sides. By drawing an analogy with the identity, we can transition from an algebraic expression to a trigonometric one, simplifying the integration process by turning the original expression into a more familiar trigonometric form.
Substitution Method
The substitution method is a technique used in integral calculus to simplify an integral, making it easier to solve. The idea is to replace a given variable or expression with a substitution that simplifies the integrand.

When we choose \(x = 6\tan\theta\) as our substitution for an expression like \(\sqrt{x^{2}+36}\), we're using the relationship between a linear expression and a trigonometric one to transform the integral. Following this substitution, we can calculate differentials and proceed with integration in terms of the new variable, \(\theta\), ultimately manipulating the integral into a more workable form.
Differential Calculus
Differential calculus is the study of how a function's value changes as its inputs change, which is encapsulated in the concept of a derivative. The derivative represents the rate of change or the slope of the function at any given point.

In the context of trigonometric substitution, differentiation is used to find the differential \(dx\) after we've made our substitution. By determining the derivative \(\frac{dx}{d\theta} = 6\sec^{2}\theta\) we obtain the differential that will replace \(dx\) in the integral, thus allowing us to express the entire integrand in terms of \(\theta\), leading to a solvable integral in the substitution domain.

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Most popular questions from this chapter

Trapezoid Rule and Simpson's Rule Consider the following integrals and the given values of \(n .\) a. Find the Trapezoid Rule approximations to the integral using \(n\) and \(2 n\) subintervals. b. Find the Simpson's Rule approximation to the integral using \(2 n\) subintervals. It is easiest to obtain Simpson's Rule approximations from the Trapezoid Rule approximations, as in Example \(8 .\) c. Compute the absolute errors in the Trapezoid Rule and Simpson's Rule with \(2 n\) subintervals. $$\int_{1}^{e} \frac{d x}{x} ; n=50$$

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