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Chapter 5: Problem 55

Evaluate the integral. $$ \int_{0}^{\sqrt{2} / 4} \frac{2}{\sqrt{1-4 x^{2}}} d x $$

Short Answer

Expert verified
The value of the given integral is \(\pi / 2\).

Step by step solution

01

Recognize the Standard Integral Form

Compare the given integral with the standard integral \(\int dx / \sqrt{1 - a^2x^2} = \sin^{-1}(ax) / a + C\). Notice that in our integral, \(a^2 = 4\), so \(a = 2\) and we have a multiplicative constant of \(2\).
02

Evaluate the Indefinite Integral

The indefinite integral will be \(\int \frac{2}{\sqrt{1-4 x^{2}}} dx = \frac{2}{2} \sin^{-1}(2x) + C = \sin^{-1}(2x) + C\).
03

Evaluate the Definite Integral

The definite integral from \(0\) to \(\sqrt{2}/4\) is evaluated by substituting these values into our expression for the indefinite integral. The result is \([\sin^{-1}(2 \cdot \sqrt{2}/4)] - [\sin^{-1}(2 \cdot 0)] = \sin^{-1}(1) - \sin^{-1}(0) = \pi / 2 - 0 = \pi / 2\).

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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 mathematical analysis that deals with the accumulation of quantities and the area under curves. Its main operation, integration, can be used for finding volumes, areas, and solving problems in physics and engineering related to rates of change.

For instance, the exercise \[\begin{equation}\int_{0}^{\sqrt{2} / 4} \frac{2}{\sqrt{1-4 x^{2}}} d x\end{equation}\]is a definite integral, which determines the area under the curve of the function \( \frac{2}{\sqrt{1-4x^2}} \) from \( x=0 \) to \( x=\frac{\sqrt{2}}{4} \). The result of this process is a number that describes the accumulation of the function's value over the given interval.
Inverse Trigonometric Functions
Inverse trigonometric functions allow us to find angles when given their trigonometric ratios. These functions include \( \arcsin(x) \), \( \arccos(x) \), and \( \arctan(x) \) among others, where \( \arcsin(x) \) is the inverse of sine.

In the context of the given exercise, the integral involves the inverse sine function \( \sin^{-1}(x) \), which appears when integrating expressions of the form \( \frac{1}{\sqrt{1-x^2}} \). The step-by-step solution makes use of this inverse trigonometric function to find the antiderivative, which leads to evaluating \( \sin^{-1}(2x) \) within the limits of integration.
Integration Techniques
Various techniques are available to solve integrals, such as substitution, integration by parts, and recognizing standard integral forms. The latter is crucial as it allows one to quickly identify an integral that matches a known result.

In this exercise, the integral matched a standard form involving the inverse sine function, which simplified the process. Understanding such techniques is important as it provides a toolkit for tackling more complicated integrals that one might encounter in calculus.
Standard Integral Forms
Memorizing standard integral forms is a key aspect of solving integrals efficiently. The form \( \int dx / \sqrt{1 - a^2x^2} = \sin^{-1}(ax) / a + C \) is one such standard result, commonly used in calculus. Recognizing this pattern in the original integral allowed the problem solver to quickly find the indefinite integral and then evaluate the definite integral.

These standard forms often involve trigonometric, exponential, logarithmic, or polynomial expressions, and mastering them can significantly reduce the complexity of integration tasks faced by students.

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Most popular questions from this chapter

Without integrating, state the integration formula you can use to integrate each of the following. (a) \(\int \frac{e^{x}}{e^{x}+1} d x\) (b) \(\int x e^{x^{2}} d x\)

Verify the differentiation formula. $$ \frac{d}{d x}\left[\sinh ^{-1} x\right]=\frac{1}{\sqrt{x^{2}+1}} $$

Let \(f\) and \(g\) be one-to-one functions. Prove that (a) \(f \circ g\) is one-to- one and (b) \((f \circ g)^{-1}(x)=\left(g^{-1} \circ f^{-1}\right)(x)\).

Use the functions \(f(x)=\frac{1}{8} x-3\) and \(g(x)=x^{3}\) to find the given value. \(\left(g^{-1} \circ f^{-1}\right)(-3)\)

Discuss several ways in which the hyperbolic functions are similar to the trigonometric functions.
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