Problem 55 - Essential Calculus

Question

Evaluate the definite integral. $$ \int_{-\pi / 2}^{\pi / 2} \cos ^{3} x d x $$

Step-by-Step Solution

Answer

The definite integral \( \int_{-\pi / 2}^{\pi / 2} \cos ^{3} x \, dx \) is equal to \( \frac{1}{2} \).

Step 1: Transform the integral

The integral of \( \cos ^{3} x \) is a bit difficult to handle directly. This exercise can be simplified by using the following power-reduction identity: \[ \cos^{3}x = \frac{1}{4}\cos(x) + \frac{3}{4}\cos(3x) \]. Substituting into the original integral gives us \[ \int_{-\pi / 2}^{\pi / 2} \cos ^{3} x \, dx = \int_{-\pi / 2}^{\pi / 2} \left(\frac{1}{4}\cos(x) + \frac{3}{4}\cos(3x)\right) \, dx \].

Step 2: Break down the integral into simpler parts

The linearity of the integral allows us to break down the integral into simpler parts which can be handled individually: \[ \int_{-\pi / 2}^{\pi / 2} \left(\frac{1}{4}\cos(x) + \frac{3}{4}\cos(3x)\right) dx = \frac{1}{4}\int_{-\pi / 2}^{\pi / 2} \cos(x) dx + \frac{3}{4}\int_{-\pi / 2}^{\pi / 2} \cos(3x) dx \]

Step 3: Calculate the integrals

Now, each integral can be calculated individually. The integral of \(\cos(x)\) is \(\sin(x)\), and the integral of \(\cos(3x)\) is \( \frac{1}{3}\sin(3x) \): \[ \frac{1}{4}\int_{-\pi / 2}^{\pi / 2} \cos(x) dx = \frac{1}{4} [\sin(x)]_{-\pi / 2}^{\pi / 2} = \frac{1}{2} \] \[ \frac{3}{4}\int_{-\pi / 2}^{\pi / 2} \cos(3x) dx = \frac{3}{4} [\frac{1}{3}\sin(3x)]_{-\pi / 2}^{\pi / 2} = 0 \]

Step 4: Combine the results

Finally, we combine the results obtained in step 3. The Integral is equal to the sum of these results: \[ \frac{1}{2} + 0 = \frac{1}{2} \].

Key Concepts

Integration Techniques

Integration techniques are methods used to solve integrals, which are fundamental in calculus for finding areas under curves. They aid in transforming and evaluating integrals that might seem challenging initially, like the definite integral of \( \cos^3 x \). Imagine them as tools that reshape the problem into something more manageable.

For integrals that involve powers of trigonometric functions, techniques such as substitution and algebraic manipulation, like breaking into sums of simpler functions, are often employed. In our exercise, the power-reduction identity is applied. The definite integral is split into simpler parts using the property of linearity of integrals. This means that:

• An integral of a sum can be separated into the sum of integrals.
• Each part is integrated individually, simplifying the overall process.

Learning these techniques increases efficiency and understanding in calculus, making seemingly complex integrations manageable.

Power-Reduction Identity

The power-reduction identity is a crucial tool in calculus, used to simplify integrals involving powers of trigonometric functions. As seen in this exercise, the identity allows us to convert \( \cos^3 x \) into a linear combination of cosine functions with different arguments. It states:

\[ \cos^{3}x = \frac{1}{4}\cos(x) + \frac{3}{4}\cos(3x) \]

This transformation simplifies the integration process. Instead of struggling with a third-degree cosine, we effectively break it down into more manageable parts. Using this identity involves recognizing a complex term and applying known formulas or identities to transform it. By reducing powers, we peel back layers, making integration or differentiation problems easier to handle. This not only provides a simpler approach to solving the problem but increases clarity and mathematical elegance.

Cosine Function Evaluation

Evaluating the cosine function involves finding the integral of cosine expressions over a specific interval. In this exercise, two main cosine functions emerge after the power-reduction: \( \cos(x) \) and \( \cos(3x) \). We find their integrals as follows:

The integral of \( \cos(x) \) is \( \sin(x) \), a direct antiderivative.

When integrating \( \cos(3x) \), we apply a direct substitution or scaling transformation, leading to the antiderivative \( \frac{1}{3}\sin(3x) \).

To evaluate these on the interval from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\), perform these steps:

• Calculate the definite integral of each function by substituting the upper and lower limits, then find the difference. This gives:
  \[ [\sin(x)]_{-\pi / 2}^{\pi / 2} \, and \, \left[ \frac{1}{3}\sin(3x) \right]_{-\pi / 2}^{\pi / 2} \]
• The integration process includes simple trigonometric calculations: the symmetry of sine and cosine helps, as their definite integrals over symmetric limits often yield a simpler format or zero.

This understanding of cosine function evaluation helps in dealing with various trigonometric integrals.