Question

Evaluate each definite integral to three significant digits. Check some by calculator. $$\int_{0}^{1} \frac{x d x}{\sqrt{4+x^{2}}}$$

Short answer

\(\frac{1}{2}\)

Step-by-step solution

Recognize the integral type

Identify the integral as an improper rational function where the variable x is in the numerator and the denominator is a square root of a sum of a constant and a square of x.

Set up the integral

Write the integral in proper notation: \[\int_{0}^{1} \frac{x dx}{\sqrt{4+x^{2}}}\]

Choose a substitution

Use trigonometric substitution to simplify the integration process. Let \( x = 2\tan(\theta) \), which implies \( dx = 2\sec^{2}(\theta) d\theta \) and \( \sqrt{4 + x^2} = \sqrt{4 + (2\tan(\theta))^2} = \sqrt{4\sec^2(\theta)} = 2\sec(\theta) \).

Change the limits of integration

Since we are using a substitution, the limits of integration also change. When \( x = 0 \), \( \theta = \arctan(\frac{x}{2}) = \arctan(0) = 0 \). When \( x = 1 \), \( \theta = \arctan(\frac{1}{2}) \).

Substitute and simplify

Substitute \( x \) and \( dx \) into the integral and simplify: \[\int_{0}^{\arctan(\frac{1}{2})} \frac{2\tan(\theta) \cdot 2\sec^{2}(\theta)}{2\sec(\theta)} d\theta = 4\int_{0}^{\arctan(\frac{1}{2})} \tan(\theta)\sec(\theta) d\theta\]

Perform the integration

Integrate the function: \[4\int_{0}^{\arctan(\frac{1}{2})} \tan(\theta)\sec(\theta) d\theta = 4\left[\frac{1}{2}\sec^2(\theta)\right]_{0}^{\arctan(\frac{1}{2})}\]

Evaluate the integrated function at the new limits

Plug in the limits of integration: \[4\left(\frac{1}{2}\sec^2(\arctan(\frac{1}{2})) - \frac{1}{2}\sec^2(0)\right)\]And remember that \(\sec(\theta) = \sqrt{1 + \tan^2(\theta)}\), thus: \[2\left(1 + (\frac{1}{2})^2\right) - 2\left(1\right) = 2\left(1 + \frac{1}{4}\right) - 2 = 2\cdot \frac{5}{4} - 2 = \frac{5}{2} - 2 = \frac{1}{2}\]

Final answer

The value of the definite integral rounded to three significant digits is: \[\int_{0}^{1} \frac{x dx}{\sqrt{4+x^{2}}} = \frac{1}{2}\]

Key concepts

Improper Rational Function

In the realm of calculus, an improper rational function is one where the numerator and denominator involve polynomials and the degree of the numerator is less than, equal to, or greater than the degree of the denominator. The function \(\frac{x}{\sqrt{4+x^2}}\) is considered 'improper' because the inclusion of the square root makes standard polynomial long division impossible.

When faced with such functions, particularly in the context of integration, we must often turn to clever methods to reshape the integral into a more manageable form, such as trigonometric substitution or partial fractions. With the function in the exercise, \(x\) is in the numerator, and the square root of a sum involving a constant term and \(x^2\) is in the denominator, which makes the direct approach to integration non-viable. As such, recognizing the type of function is just the first step; applying a suitable technique to solve the integral is essential.

Trigonometric Substitution

To tackle integrals with square roots involving sum or difference of squares, such as \(\sqrt{a^2 + x^2}\), \(\sqrt{a^2 - x^2}\), or \(\sqrt{x^2 - a^2}\), trigonometric substitution is a powerful tool. It simplifies the integral by using the properties of trigonometric functions that naturally contain roots in their identities, such as the Pythagorean identity \(1 + \tan^2(\theta) = \sec^2(\theta)\).

In this exercise, the substitution \(x = 2\tan(\theta)\) is used to rewrite the integral and eliminate the square root. This changes the integral from one involving \(x\) to one involving \(\theta\), which is easier to integrate. This technique is reliant on our knowledge of trigonometric identities and is only applicable when the integral conforms to patterns similar to trigonometric expressions. After substitution, the integral often simplifies to a standard trigonometric integral, which is much more straightforward to solve.

Integration Limits

Understanding integration limits is crucial when performing definite integrals. The limits indicate the range over which we are integrating, usually corresponding to the area under the curve from one point to another on the x-axis.

When a substitution is made, as with trigonometric substitution in our example, we need to change the integration limits to match the substituted variable. This step is vital because the original limits pertain to the \(x\)-values, and after substitution, we're integrating with respect to \(\theta\). The new limits of integration are found by plugging the original \(x\) limits into the substitution equation.

For instance, if you substitute \(x = 2\tan(\theta)\), when \(x = 0\), \(\theta\) will be \(\arctan(0)\) which is 0. When \(x = 1\), \(\theta\) will be \(\arctan(\frac{1}{2})\). These new limits ensure the definite integral is evaluated over the correct interval in terms of \(\theta\) instead of \(x\), allowing for accurate computation.