Evaluate The Integral:${ \int \cos \left(6x - \frac{\pi}{2}\right) , Dx }$A. { \frac{1}{6} \cos \left(6x - \frac{\pi}{2}\right) + C$}$B. { \frac{1}{6} \sin \left(6x - \frac{\pi}{2}\right) + C$} C . \[ C. \[ C . \[ -\frac{1}{6}

Mar 10, 2025 by ADMIN 228 views

**Evaluating Trigonometric Integrals: A Step-by-Step Guide**

Introduction

In calculus, integrals are used to find the area under curves and are a fundamental concept in mathematics. When dealing with trigonometric functions, evaluating integrals can be a bit more challenging. In this article, we will focus on evaluating the integral of a cosine function with a specific argument. We will break down the problem step by step and provide a clear explanation of the solution.

The Integral to be Evaluated

The integral we want to evaluate is:

\int \cos \left(6x - \frac{\pi}{2}\right) \, dx

Step 1: Identify the Trigonometric Identity

To evaluate this integral, we need to recognize a trigonometric identity that will help us simplify the expression. In this case, we can use the identity:

\cos(A - B) = \cos A \cos B + \sin A \sin B

However, in this problem, we can use a different approach. We can use the identity:

\cos(A - B) = \cos A \cos B + \sin A \sin B

But we can also use the identity:

\cos(A + B) = \cos A \cos B - \sin A \sin B

We can rewrite the integral as:

\int \cos \left(6x - \frac{\pi}{2}\right) \, dx = \int \cos(6x) \cos\left(\frac{\pi}{2}\right) - \sin(6x) \sin\left(\frac{\pi}{2}\right) \, dx

Step 2: Simplify the Expression

Using the values of $\cos\left(\frac{\pi}{2}\right) = 0$ and $\sin\left(\frac{\pi}{2}\right) = 1$ , we can simplify the expression:

\int \cos \left(6x - \frac{\pi}{2}\right) \, dx = \int -\sin(6x) \, dx

Step 3: Evaluate the Integral

Now we can evaluate the integral:

\int -\sin(6x) \, dx = \frac{1}{6} \cos(6x) + C

Conclusion

In conclusion, the integral of $\cos \left(6x - \frac{\pi}{2}\right)$ is $\frac{1}{6} \cos \left(6x - \frac{\pi}{2}\right) + C$ . This is option A.

Comparison with Other Options

Let's compare our answer with the other options:

Option B: $\frac{1}{6} \sin \left(6x - \frac{\pi}{2}\right) + C$
Option C: $-\frac{1}{6} \cos \left(6x - \frac{\pi}{2}\right) + C$

Our answer is different from option B, which is incorrect. Our answer is also different from option C, which is also incorrect.

Discussion

In this article, we evaluated the integral of a cosine function with a specific argument. We used trigonometric identities to simplify the expression and then evaluated the integral. We compared our answer with other options and found that our answer is correct.

Final Answer

The final answer is:

\boxed{\frac{1}{6} \cos \left(6x - \frac{\pi}{2}\right) + C}

Additional Resources

For more information on trigonometric integrals, please refer to the following resources:

Conclusion

Introduction

In our previous article, we evaluated the integral of a cosine function with a specific argument. We used trigonometric identities to simplify the expression and then evaluated the integral. In this article, we will provide a Q&A guide to help you understand the concepts and techniques used in evaluating trigonometric integrals.

Q: What is a trigonometric integral?

A: A trigonometric integral is an integral that involves a trigonometric function, such as sine or cosine, as the integrand.

Q: What are some common trigonometric identities used in evaluating integrals?

A: Some common trigonometric identities used in evaluating integrals include:

$\cos(A - B) = \cos A \cos B + \sin A \sin B$
$\cos(A + B) = \cos A \cos B - \sin A \sin B$
$\sin(A - B) = \sin A \cos B - \cos A \sin B$
$\sin(A + B) = \sin A \cos B + \cos A \sin B$

Q: How do I evaluate the integral of a cosine function with a specific argument?

A: To evaluate the integral of a cosine function with a specific argument, you can use the following steps:

Identify the trigonometric identity that will help you simplify the expression.
Simplify the expression using the trigonometric identity.
Evaluate the integral using the simplified expression.

Q: What is the difference between the integral of a cosine function and the integral of a sine function?

A: The integral of a cosine function is typically of the form:

\int \cos(ax + b) \, dx

where $a$ and $b$ are constants.

The integral of a sine function is typically of the form:

\int \sin(ax + b) \, dx

where $a$ and $b$ are constants.

Q: How do I evaluate the integral of a sine function with a specific argument?

A: To evaluate the integral of a sine function with a specific argument, you can use the following steps:

Identify the trigonometric identity that will help you simplify the expression.
Simplify the expression using the trigonometric identity.
Evaluate the integral using the simplified expression.

Q: What are some common mistakes to avoid when evaluating trigonometric integrals?

A: Some common mistakes to avoid when evaluating trigonometric integrals include:

Not identifying the correct trigonometric identity to use.
Not simplifying the expression correctly.
Not evaluating the integral correctly.

Q: How do I check my answer for a trigonometric integral?

A: To check your answer for a trigonometric integral, you can use the following steps:

Plug in the original expression into the integral.
Simplify the expression using the trigonometric identity.
Evaluate the integral using the simplified expression.
Compare your answer with the original expression.

Conclusion

In conclusion, evaluating trigonometric integrals requires a good understanding of trigonometric identities and integration techniques. By following the steps outlined in this article, you can evaluate integrals of cosine and sine functions with specific arguments. Remember to identify the correct trigonometric identity to use, simplify the expression correctly, and evaluate the integral correctly.

Additional Resources

For more information on trigonometric integrals, please refer to the following resources:

Final Answer

The final answer is:

\boxed{\frac{1}{6} \cos \left(6x - \frac{\pi}{2}\right) + C}