Problem 37 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(33-38\). $$ \int_{0}^{\pi} \cos 3 x \cos 4 x d x $$

Step-by-Step Solution

Answer

The integral evaluates to 0.

Step 1: Use the Product-to-Sum Formula

To solve this integral, we will use the product-to-sum identities from trigonometry. The formula is: \[\cos A \cos B = \frac{1}{2} \left( \cos(A + B) + \cos(A - B) \right)\]Applying this to our integral, we have:\[\cos 3x \cos 4x = \frac{1}{2} \left( \cos(3x + 4x) + \cos(3x - 4x) \right) = \frac{1}{2} \left( \cos 7x + \cos(-x) \right)\]Since \(\cos(-x) = \cos x\), this becomes:\[\frac{1}{2} \left( \cos 7x + \cos x \right)\]

Step 2: Split the Integral Into Two Parts

Now, substitute the expression from Step 1 into the integral and split the integral as follows:\[\int_{0}^{\pi} \cos 3x \cos 4x \, dx = \int_{0}^{\pi} \frac{1}{2} \left( \cos 7x + \cos x \right) \, dx\]This can be rewritten as:\[\frac{1}{2} \left( \int_{0}^{\pi} \cos 7x \, dx + \int_{0}^{\pi} \cos x \, dx \right)\]

Step 3: Evaluate the Integrals Separately

We evaluate each integral separately.For \(\int_{0}^{\pi} \cos 7x \, dx \):\[\int \cos 7x \, dx = \frac{1}{7} \sin 7x + C\]Thus, the definite integral from 0 to \(\pi\) is:\[\left[ \frac{1}{7} \sin 7x \right]_{0}^{\pi} = \frac{1}{7} (\sin 7\pi - \sin 0) = 0\]For \(\int_{0}^{\pi} \cos x \, dx \):\[\int \cos x \, dx = \sin x + C\]Thus, the definite integral from 0 to \(\pi\) is:\[\left[ \sin x \right]_{0}^{\pi} = \sin \pi - \sin 0 = 0\]

Step 4: Combine the Results

Combine the results of each integral back into the original expression:\[\frac{1}{2} \left( 0 + 0 \right) = 0\]Thus, the value of the given integral \(\int_{0}^{\pi} \cos 3x \cos 4x \, dx\) is 0.

Key Concepts

Product-to-Sum Formulas

Trigonometric identities provide a set of formulas that simplify the multiplication of trigonometric functions, such as sine and cosine, into sums or differences. The product-to-sum formulas are especially useful when dealing with integrals that involve the product of two trigonometric functions.

In the case of the given integral, \(\int_{0}^{\pi} \cos 3x \cos 4x \, dx\), the product-to-sum formula becomes essential to simplify the integrand. The product-to-sum formula for cosine is:

• \(\cos A \cos B = \frac{1}{2} (\cos(A + B) + \cos(A - B))\)

To apply this to our integral, we set \(A = 3x\) and \(B = 4x\). The formula then transforms the initial product into a sum, making the mathematical analysis and integration much easier. Using trigonometric identities not only simplifies calculations but can also turn complex problems into straightforward exercises.

Definite Integrals

A definite integral represents the signed area under a curve within a specific interval on the x-axis. In simpler terms, when you compute a definite integral, you find the net area between the curve and the x-axis from the lower limit to the upper limit of integration.

In our exercise, the definite integral is evaluated from 0 to \( \pi \). This involves calculating the difference of the antiderivative at these boundary points. For any function \(f(x)\), the definite integral from \(a\) to \(b\) can be described as:

• \[\int_{a}^{b} f(x) \, dx = F(b) - F(a)\]

Here, \(F(x)\) represents the antiderivative of \(f(x)\). The process involves finding this antiderivative, evaluating it at both the upper and lower limits of integration, and then taking their difference. With trigonometric functions, definite integrals often yield zero due to the periodic nature over certain symmetric limits, such as 0 and \(\pi\).

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables involved. These identities are crucial not only for solving integrals but also for simplifying expressions in mathematics and engineering.

Some of the key trigonometric identities include the Pythagorean identity, angle sum and difference formulas, and the product-to-sum identities. For instance, knowing that \(\cos(-x) = \cos x\) helps simplify expressions when working with integrals involving cosine. These identities allow us to transform and simplify trigonometric expressions, making complex integrals more approachable.
Here are a few fundamental trigonometric identities:

• \(\sin^{2} \theta + \cos^{2} \theta = 1\)
• \(\cos(a + b) = \cos a \cos b - \sin a \sin b\)
• \(\cos(A + B) + \cos(A - B) = 2\cos A\cos B\)

Utilizing these identities in calculus, particularly within integrals, helps streamline the problem-solving process by reducing complicated trigonometric functions into simpler forms that are easier to integrate.