Problem 432 - Calculus Volume 1

Question

In the following exercises, compute each definite integral. $$\int_{0}^{1 / 2} \frac{\cos \left(\tan ^{-1} t\right)}{1+t^{2}} d t$$

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{1}{2} \tan^{-1}(1/2) \).

Step 1: Recognize the Function

Observe that the given integral is \( \int_{0}^{1 / 2} \frac{\cos(\tan^{-1} t)}{1+t^2} \, dt \). We need to simplify the expression \( \frac{\cos(\tan^{-1} t)}{1+t^2} \).

Step 2: Use Trigonometric Identity

Recall that for an angle \( \theta = \tan^{-1}(t) \), \( t = \tan(\theta) \). Using the identity \( 1 + \tan^2(\theta) = \sec^2(\theta) \), we have \( \cos(\tan^{-1}(t)) = \frac{1}{\sec(\theta)} = \cos(\theta) \), and \( 1 + t^2 = \sec^2(\theta) \). This simplifies \( \frac{\cos(\tan^{-1} t)}{1 + t^2} \) to \( \cos(\theta) / \sec^2(\theta) = \cos^3(\theta) \).

Step 3: Simplify the Integrand

Using the previous identity simplifications, the integrand \( \frac{\cos(\tan^{-1} t)}{1 + t^2} \) becomes \( \cos^2(\tan^{-1} t) \). Thus, the integral becomes \( \int_{0}^{1 / 2} \cos^2(\tan^{-1} t) \, dt \).

Step 4: Variable Substitution

Perform a substitution: let \( \theta = \tan^{-1}(t) \) so that \( d\theta = \frac{1}{1+t^2} \, dt \). When \( t = 0 \), \( \theta = 0 \), and when \( t = \frac{1}{2} \), \( \theta = \tan^{-1}(\frac{1}{2}) \). Thus, the integral becomes \( \int_{0}^{\tan^{-1}(1/2)} \cos^2(\theta) \, d\theta \).

Step 5: Use Integration Formula for Cosine Squared

Use the trigonometric identity \( \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \). The integral becomes \( \frac{1}{2} \int_{0}^{\tan^{-1}(1/2)} (1 + \cos(2\theta)) \, d\theta \).

Step 6: Calculate the Integral

Separate the integral into two parts: \( \frac{1}{2} \left[ \int_{0}^{\tan^{-1}(1/2)} 1 \, d\theta + \int_{0}^{\tan^{-1}(1/2)} \cos(2\theta) \, d\theta \right] \). This evaluates to:\( \frac{1}{2} \left[ \theta + \frac{1}{2} \sin(2\theta) \right]_{0}^{\tan^{-1}(1/2)} \). Since \( \sin(2\theta) \) evaluated at the limits is zero, the main part becomes \( \frac{1}{2} [ \theta ]_{0}^{\tan^{-1}(1/2)} = \frac{1}{2} \tan^{-1}(1/2) \).

Step 7: Final Answer

The definite integral evaluates to \( \frac{1}{2} \tan^{-1}(1/2) \), which is the value of the integral over the bounds given.

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical equations that relate various angles and trigonometric functions such as sine, cosine, and tangent. In calculus, they play a crucial role in simplifying complex expressions and making integration more approachable. For example, consider the expression \( \cos(\tan^{-1}(t)) \). By recognizing that \( \theta = \tan^{-1}(t) \), we can use the identity \( 1 + \tan^2(\theta) = \sec^2(\theta) \). Based on this, the cosine function can be simplified: \( \cos(\theta) = \frac{1}{\sec(\theta)} \).

• This simple identity helps in rewriting difficult integrands into manageable forms.
• Reduction often uses known identities like \( \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \).

These identities are fundamental, especially when integrating trigonometric functions, allowing us to progress with seemingly unsolvable integrals.

Integration by Substitution

Integration by substitution is an essential technique in calculus, especially useful for solving integrals that aren't straightforward. This method involves changing variables to transform a hard-to-solve integral into a simpler one. In our exercise, we perform a substitution where \( \theta = \tan^{-1}(t) \). Here's why this is effective:

• It aligns the integral limits with the new variable. When \( t = 0 \), \( \theta = 0 \), and when \( t = \frac{1}{2} \), \( \theta = \tan^{-1} \left( \frac{1}{2} \right) \).
• The differential \( dt \) is replaced using \( d\theta = \frac{1}{1 + t^2} \, dt \), which simplifies integration to \( \int \cos^2(\theta) \, d\theta \).

This substitution simplifies calculations and takes advantage of easier integration facts, leading to a more manageable solution. Such transformations are pivotal when dealing with trigonometric or inverse trigonometric functions.

Cosine Function

The cosine function is a fundamental part of trigonometry, representing the x-coordinate of a point on the unit circle. When dealing with integrals, the properties of the cosine function can be leveraged for simplification, especially with identities. In this problem, upon transforming the integral to \( \int \cos^2(\theta) \, d\theta \), it becomes manageable due to its trigonometric property:

• Using \( \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \), the integrand separates into basic components.
• This identity reduces the complexity, allowing integration over well-known functions like constants and cosine multiples.

The use of the cosine function extends beyond simple numerical calculations to aiding in effective simplifications with its rich set of identities. Its periodic nature and connection to angles offer multiple routes for tackling integration. Through identities and substitution, recognizing and transforming the cosine function greatly assists in evaluating definite integrals.