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Chapter 8: Problem 1

Evaluate the integrals. $$\int \frac{d x}{\sqrt{9+x^{2}}}$$

Short Answer

Expert verified
\( \sinh^{-1}(\frac{x}{3}) + C \)

Step by step solution

01

Identify the Integral Form

The given integral \( \int \frac{d x}{\sqrt{9+x^{2}}} \) is of the form \( \int \frac{1}{\sqrt{a^2 + x^2}} \, dx \), which is a standard integral form.
02

Recognize the Trigonometric Identity

For the integral \( \int \frac{d x}{\sqrt{a^2 + x^{2}}} \), we recognize that it can be solved using the formula \( \sinh^{-1}(\frac{x}{a}) + C \), where \( a = 3 \) in this case.
03

Apply the Formula

Using the identified formula, \( \int \frac{d x}{\sqrt{9+x^{2}}} = \sinh^{-1}(\frac{x}{3}) + C \), where \( C \) is the constant of integration.
04

Write the Final Answer

The integral evaluates to \( \sinh^{-1}(\frac{x}{3}) + C \).

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Key Concepts

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

Trigonometric Identities
Trigonometric identities are fundamental tools in calculus, especially when dealing with integrals. They allow us to simplify complex expressions. In this context, identities like the Pythagorean identity are extremely useful. They help us recognize patterns in integrals that might not be immediately obvious. Remember the Pythagorean identity:
  • \( \sin^2(\theta) + \cos^2(\theta) = 1 \)
This identity and others allow us to transform integrals into forms that are easier to integrate. In the given exercise, although we initially recognize a hyperbolic function, understanding trigonometric identities is key to transforming and simplifying
expressions involving roots and squares.
Standard Integral Forms
In calculus, a number of standard integral forms help us to quickly solve integrals. These are known as reference integrals and they cover common patterns and scenarios you might encounter. Observing the integral \( \int \frac{1}{\sqrt{a^2 + x^2}} \, dx \), we can immediately identify its solution using a standard form related to the inverse hyperbolic sine function. These standard forms act like a toolkit in integration, enabling solutions without lengthy
calculations:
  • \( \int \frac{1}{\sqrt{a^2 + x^2}} \, dx = \sinh^{-1}(\frac{x}{a}) + C \)
  • These forms often involve logarithmic and trigonometric functions.
Recognizing these patterns simplifies our work and reduces the chances of errors.
Hyperbolic Functions
Hyperbolic functions resemble trigonometric functions, with similar names and properties:
  • \( \sinh(x) \)
  • \( \cosh(x) \)
  • Hyperbolic functions are defined using exponential functions.
The integral in our exercise involved hyperbolic functions: specifically the inverse hyperbolic sine, \( \sinh^{-1}(x) \). This function provides a solution to the integral involving \( \int \frac{1}{\sqrt{a^2 + x^2}} \, dx \). Hyperbolic identities help further in transforming and simplifying expressions, similar to trigonometric identities. It's important to understand their properties as they are frequently used in engineering and physics
applications where their roles are as critical as their trigonometric counterparts.

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Most popular questions from this chapter

Solve the initial value problems in Exercises \(51-54\) for \(x\) as a function of \(t\). $$\left(3 t^{4}+4 t^{2}+1\right) \frac{d x}{d t}=2 \sqrt{3}, \quad x(1)=-\pi \sqrt{3} / 4$$

The integral $$\mathrm{Si}(x)=\int_{0}^{x} \frac{\sin t}{t} d t$$ called the sine-integral function, has important applications in optics. a. Plot the integrand \((\sin t) / t\) for \(t>0 .\) Is the sine-integral function everywhere increasing or decreasing? Do you think Si \((x)=0\) for \(x>0 ?\) Check your answers by graphing the function \(\operatorname{Si}(x)\) for \(0 \leq x \leq 25.\) b. Explore the convergence of $$\int_{0}^{\infty} \frac{\sin t}{t} d t.$$ If it converges, what is its value?

Find the values of \(p\) for which each integral converges. a. \(\int_{1}^{2} \frac{d x}{x(\ln x)^{p}}\) b. \(\int_{2}^{\infty} \frac{d x}{x(\ln x)^{p}}\)

In each case, check your work by differentiating your answer with respect to \(x\). $$\int \tanh ^{-1} x d x$$

In Exercises \(33-38,\) perform long division on the integrand, write the proper fraction as a sum of partial fractions, and then evaluate the integral. $$\int \frac{y^{4}+y^{2}-1}{y^{3}+y} d y$$
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