Question

True or False. Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ 4 \int \sin x \cos x d x=-\cos 2 x+C $$

Short answer

The statement is false. The actual integral of \(4 \sin x \cos x dx\) is \(-2 \cos 2x + C\), not \(-\cos 2x + C\). The discrepancy is due to the coefficient '4' at the beginning of the integral.

Step-by-step solution

Understand given equation

In the given statement, there's an integral equation. This equation states that the integral of \(4 \sin x \cos x\) equals to \(- \cos 2x + C\), where \(C\) is the constant of integration. This is a case of integral of product of sines and cosines.

Evaluate given integral

The integral of \(4 \sin x \cos x\) can be evaluated using the power-reducing identity, which turns the product of \(\sin x\) and \(\cos x\) into a double angle formula. In this case, \(2\sin x \cos x = \sin 2x\). Therefore, the integral becomes \(\int 2 \sin 2x dx = -\cos 2x + C_1\), where \(C_1\) is a constant.

Compare results

Now, comparing the calculated integral, \(-\cos 2x + C_1\), with the one given in the statement, \(-\cos 2x + C\). We can see these results are not the same because the coefficient in the integral calculated is '2' and not '4', as it is in the given equation. Even when we allow free constant of integration.

Key concepts

Trigonometric Integrals

Understanding the integration of trigonometric functions is crucial for solving calculus problems. Trigonometric integrals involve the integration of sin(x), cos(x), tan(x), cot(x), sec(x), and csc(x), along with their various powers and products.

The integral of the product of sin(x) and cos(x) poses a unique challenge because sin(x) and cos(x) have different properties when integrated. What's essential is recognizing patterns that allow for simplifications, such as power-reducing identities and double angle formulas, which greatly facilitate the integration process.

When confronted with an integral involving products of sine and cosine, such as \( 4 \text{sin}(x) \text{cos}(x) \), a common technique is to use trigonometric identities to transform the integrand into a more manageable form for integration.

Power-Reducing Identity

Power-reducing identities are a type of trigonometric identity that express powers of sine and cosine in terms of first powers with double or multiple angles. These are particularly useful in integration when the integrand involves higher powers of sine or cosine.

In our task, the power-reducing identity comes into play to convert the product of \( \text{sin}(x) \) and \( \text{cos}(x) \) into a double angle format. Specifically, we use the identity \( 2\text{sin}(x)\text{cos}(x) = \text{sin}(2x) \), which is directly applicable to our integrand, except for a factor of 2. This identity simplifies the integral and allows us to integrate directly, without resorting to more complex integration techniques.

Double Angle Formulas

Double angle formulas are a subset of trigonometric identities that provide relations for the sine, cosine, or tangent of twice a certain angle in terms of trigonometric functions of the original angle. For example, \( \text{sin}(2x) \) can be represented as \( 2\text{sin}(x)\text{cos}(x) \).

These formulas are incredibly useful for integrating products like \( \text{sin}(x)\text{cos}(x) \) as they transform the product into a single trigonometric function of a double angle. This can then be integrated more straightforwardly. However, in our exercise, the double angle formula needs to be correctly scaled to account for the coefficient of '4' in order to match the given equation. This oversight in scaling would result in an incorrect statement.

Integration Techniques

There are numerous integration techniques that cater to different types of functions and their products. Techniques such as substitution, integration by parts, trigonometric substitution, and partial fraction decomposition are commonly used to solve integrals that are not readily integrable.

In the case of the product of \( \text{sin}(x) \) and \( \text{cos}(x) \), we applied a specific trigonometric identity to simplify the integrand before integrating. This is a preferred technique whenever trigonometric integrals are encountered, as it often leads to a much simpler integral that can be solved directly. It's crucial, however, that students pay attention to the coefficients and constants during the simplification and integration processes to ensure the result accurately matches what is expected, as exemplified by the exercise at hand.