Problem 24 - Biocalculus Calculus for the Life Sciences

Question

Evaluate the integral. \(\int_{0}^{\pi / 3} \frac{\sin \theta+\sin \theta \tan ^{2} \theta}{\sec ^{2} \theta} d \theta\)

Step-by-Step Solution

Answer

The integral evaluates to \(\frac{1}{2}\).

Step 1: Simplify the Integrand

The given integral can be simplified by rewriting the integrand. Recall that \( \sec^2 \theta = \frac{1}{\cos^2 \theta} \), which allows us to rewrite the integrand as \( \int \frac{\sin \theta + \sin \theta \tan^2 \theta}{\sec^2 \theta} d\theta = \int (\sin \theta \cos^2 \theta + \sin \theta \sin^2 \theta) d\theta \).

Step 2: Factor out the Common Term

Factor out \( \sin \theta \) from the expression: \( \int \sin \theta (\cos^2 \theta + \sin^2 \theta) d\theta \). Since \( \cos^2 \theta + \sin^2 \theta = 1 \), the expression simplifies to \( \int \sin \theta d\theta \).

Step 3: Find the Antiderivative

The antiderivative of \( \sin \theta \) is \( -\cos \theta \). Therefore, \( \int \sin \theta d\theta = -\cos \theta + C \), where \( C \) is the constant of integration.

Step 4: Evaluate the Definite Integral

Now, evaluate the definite integral from \( 0 \) to \( \frac{\pi}{3} \):\[-\cos \theta \bigg|_{0}^{\pi/3} = -\cos(\frac{\pi}{3}) - (-\cos(0))\]Since \( \cos(\frac{\pi}{3}) = \frac{1}{2} \) and \( \cos(0) = 1 \), the expression becomes:\[-\left(\frac{1}{2} - 1\right) = 1 - \frac{1}{2} = \frac{1}{2}\]

Key Concepts

Definite Integrals

Definite integrals are an essential concept in calculus, providing a way to calculate the exact area under a curve between two points on the x-axis. This area represents the accumulation of values of a function from one point to another.
In our exercise, the goal was to evaluate a definite integral from 0 to \( \frac{\pi}{3} \). While indefinite integrals require us to add a constant \( C \), definite integrals do not, as we evaluate the exact limit points.
To evaluate a definite integral, follow these general steps:

• Find an antiderivative (or indefinite integral) of the function, which provides the function whose derivative is given.
• Use the Fundamental Theorem of Calculus, which involves subtracting the value of the antiderivative at the lower limit from the value at the upper limit.
• Substitute and simplify to get the exact numerical value.

Understanding definite integrals is crucial for solving many real-world problems, such as calculating distances and areas.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables involved. They are fundamental tools for simplifying complex trigonometric expressions, making calculus operations more manageable.
One crucial identity used in the original exercise is \( \sec^2\theta = \frac{1}{\cos^2\theta} \). This identity allows for transformations between functions that can simplify expressions.
Another critical identity is the Pythagorean identity: \( \cos^2\theta + \sin^2\theta = 1 \). By using this identity, the integrand simplifies dramatically. This step turns a complicated expression into an easy function to integrate.
Here are some commonly used trigonometric identities:

• Pythagorean identities: \( \sin^2\theta + \cos^2\theta = 1 \)
• Reciprocal identities: \( \tan\theta = \frac{\sin\theta}{\cos\theta} \)
• Angle sum and difference identities: \( \sin(a ± b) = \sin a \cos b ± \cos a \sin b \)

Grasping these identities helps you maneuver through complex integrals and algebraic manipulations efficiently.

Antiderivatives

An antiderivative, also known as an indefinite integral, is a function whose derivative is the original function. Finding antiderivatives is essential for solving integration problems as it provides the basis for calculating definite integrals.
In the problem, the antiderivative of \( \sin \theta \) is \( -\cos \theta \). This specific function is relatively simple to differentiate, making it an ideal example for understanding the concept.
General strategy for finding antiderivatives involves:

• Recognizing standard integral forms, such as \( \int \sin x \,dx = -\cos x + C \)
• Applying substitution methods, when necessary, to simplify the integrand.
• Using integration by parts or partial fraction decomposition for more complex functions.

Mastering antiderivatives aids in transitioning to more advanced mathematical problems beyond basic calculus, including solving differential equations and evaluating complex integrals.