Problem 67 - Calculus Early Transcendentals

Question

Integrals of the form \(\int \sin m x \cos n x d x\) Use the following three identities to evaluate the given integrals. $$\begin{aligned}&\sin m x \sin n x=\frac{1}{2}(\cos ((m-n) x)-\cos ((m+n) x))\\\&\sin m x \cos n x=\frac{1}{2}(\sin ((m-n) x)+\sin ((m+n) x))\\\&\cos m x \cos n x=\frac{1}{2}(\cos ((m-n) x)+\cos ((m+n) x))\end{aligned}$$ $$\int \sin 3 x \cos 7 x d x$$

Step-by-Step Solution

Answer

Answer: The integral of sin(3x)cos(7x) with respect to x is given by the expression \(\frac{1}{2} \left( -\frac{1}{4}\cos(-4x) -\frac{1}{10}\cos(10x) \right) + C\), where C is the constant of integration.

Step 1: Use the identity to rewrite the integral

We will use the second identity with \(m = 3\) and \(n = 7 \): $$\sin 3x \cos 7x = \frac{1}{2}(\sin( (3-7)x) + \sin( (3+7)x )) = \frac{1}{2}(\sin(-4x) + \sin(10x))$$ Now, the integral becomes: $$\int \sin 3x \cos 7x dx = \frac{1}{2} \int (\sin(-4x) + \sin(10x)) dx$$

Step 2: Integrate each sine function separately

Now, we will integrate each sine function inside the integral: $$\frac{1}{2} \int (\sin(-4x) + \sin(10x)) dx = \frac{1}{2} \left( \int \sin(-4x) dx + \int \sin(10x) dx \right)$$ Recall that the integral of \(\sin(kx)\) is \(-\frac{1}{k}\cos(kx)\). So, $$\frac{1}{2} \left( \int \sin(-4x) dx + \int \sin(10x) dx \right) = \frac{1}{2} \left( -\frac{1}{4}\cos(-4x) -\frac{1}{10}\cos(10x) \right) + C$$

Step 3: Write the final answer

The final answer is: $$\int \sin 3x \cos 7x dx = \frac{1}{2} \left( -\frac{1}{4}\cos(-4x) -\frac{1}{10}\cos(10x) \right) + C$$

Key Concepts

Trigonometric Identities

Understanding trigonometric identities is essential when working with integrals involving trigonometric functions. These identities are mathematical equations that express one trigonometric function in terms of others. A common example is the Pythagorean identity, which relates the square of sine and cosine functions: \[ \text{sin}^2(x) + \text{cos}^2(x) = 1 \text{.} \]
Trigonometric identities greatly simplify the process of integrating complex trigonometric expressions by transforming them into more manageable forms. As seen in the exercise, trigonometric identities for the product of sine and cosine functions can be particularly useful. For instance, \[ \text{sin}(mx) \text{cos}(nx) = \frac{1}{2}( \text{sin}((m - n)x) + \text{sin}((m + n)x) ) \text{,} \]
is used to break down the product into a sum of two separate sine functions, each of which can be integrated independently.

Definite Integral

The concept of a definite integral refers to the calculation of the area under a curve, between two specific points on the x-axis. Represented symbolically as \[ \text{the integral from } a \text{ to } b \text{ of } f(x) dx \text{,} \]
it provides a clear notion of the accumulation of quantities, such as distance or probability. In practice, definite integrals are evaluated by finding the antiderivative of the function and then applying the limits of integration. This exercise focuses on indefinite integration, which is related but finds the antiderivative without specific limits. Nonetheless, understanding definite integrals helps to ground the concept of finding areas and serves as an extension of the technique employed in indefinite integration.

Sine and Cosine

The functions sine and cosine are fundamental to trigonometry and form the basis of many trigonometric identities. These periodic functions have a range of applications from describing wave patterns to modeling circular motion. When working with integrals involving sine and cosine, you should be aware of their basic properties such as their periodicity, symmetry, and their relationship to one another through phase shifts. For example, \[ \text{sin}(x) \text{ is similar to } \text{cos}(x - \frac{\pi}{2}) \text{.} \]
The exercise provided uses identities where these functions are combined, demonstrating the interplay between them during integration processes. Understanding their graphs and behavior can simplify the task of integrating these functions over specific intervals.

Integration Techniques

There are various integration techniques that can be employed depending on the complexity of the function being integrated. Common techniques include substitution, integration by parts, partial fraction decomposition, and trigonometric substitution. The given exercise showcases an approach where a trigonometric identity simplifies the integrand into a form that is more straightforward to integrate. Once the identity is applied, the standard integration method for sine, which involves knowing that \[ \text{the integral of } \text{sin}(kx) dx \text{ is } -\frac{1}{k}\text{cos}(kx) \text{ plus a constant } C \text{,} \]
is utilized. By mastering different techniques, you can tackle a wide range of integrals, making it a key skill set for students studying calculus.