91Ó°ÊÓ

When dealing with the product of trigonometric functions, transforming them using product-to-sum identities can simplify integration. These identities turn products into sums or differences, making them easier to handle mathematically. In particular, the identity for the product of two cosine functions is very useful:

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

Applying this to our problem, where \(A = 3x\) and \(B = 4x\), we transform \(\cos 3x \cos 4x\) into a sum of cosines:

• \(\cos 3x \cos 4x = \frac{1}{2} [\cos 7x + \cos x]\)

This transformation helps in breaking down a complex integral into more manageable parts by reducing the multiplication of cosines to a simpler addition process. Understanding and mastering these identities is crucial for simplifying many trigonometric integrations.
Product-to-sum identities not only simplify computations but also provide deeper insights into the relationships between trigonometric functions and their applications in calculus.

The cosine function, \(\cos x\), is one of the most fundamental trigonometric functions. It defines the cosine of an angle \(x\), which is the adjacent side over the hypotenuse in a right triangle setting. Cosine functions repeat their values in a periodic manner, which makes them especially useful in describing oscillations, such as waves.
Its basic properties include:

• Periodicity: Repeats every \(2\pi\).
• Even Function: \(\cos(-x) = \cos x\).
• Range: Values lie between -1 and 1.

In the context of integration, the cosine function often appears with altered arguments like \(7x\) or \(3x + 4x\), corresponding to transformations seen in product-to-sum identities.
Understanding how to integrate cosine directly, as in \(\int \cos kx \, dx\), involves knowing antiderivatives, which are essential skills in calculus.

Integrating Cosine Functions

The integral of a cosine function, \(\int \cos kx \, dx\), is typically:

• \(\frac{1}{k} \sin kx + C\)

This integration rule is crucial for solving problems involving harmonic motion or wave analysis, where the cosine function models periodic phenomena.

Indefinite integration is a core concept in calculus that represents the reverse process of differentiation. It involves finding a function, called the antiderivative, whose derivative is the given function. The result of an indefinite integral includes a constant of integration, \(C\), because differentiation of a constant yields zero.
In our specific problem:

• We're integrating \(\cos 3x \cos 4x\) after applying a product-to-sum identity.

This process demonstrates how indefinite integration requires breaking down complex expressions into smaller parts. By expressing \(\cos 3x \cos 4x\) using the identity:

• \(\cos 3x \cos 4x = \frac{1}{2} [\cos 7x + \cos x]\)

we can separately integrate each term:

• \(\int \cos 7x \, dx = \frac{1}{7} \sin 7x + C\)
• \(\int \cos x \, dx = \sin x + C\)

Finally, these are combined to form the complete indefinite integrals solution. Integrating in this manner emphasizes the importance of recognizing patterns and identities that simplify calculations.