Evaluate The Integral:$\[ +\frac{9}{2} \int \frac{1}{\sec \theta} \tan \theta \, D\theta \\]

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Introduction

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In this article, we will delve into the world of calculus and evaluate the given integral. The integral in question is $\int \frac{1}{\sec \theta} \tan \theta \, d\theta$. We will break down the solution step by step, using various mathematical techniques and formulas to simplify the expression and arrive at the final answer.

Understanding the Integral

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Before we begin, let's take a closer look at the integral. The integral is $\int \frac{1}{\sec \theta} \tan \theta \, d\theta$. We can see that the integrand is a product of two trigonometric functions: $\frac{1}{\sec \theta}$ and $\tan \theta$. The secant function is the reciprocal of the cosine function, and the tangent function is the ratio of the sine function to the cosine function.

Step 1: Simplify the Integrand

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To simplify the integrand, we can use the trigonometric identity $\sec \theta = \frac{1}{\cos \theta}$. Substituting this into the integrand, we get:

$$\frac{1}{\sec \theta} \tan \theta = \frac{1}{\frac{1}{\cos \theta}} \tan \theta = \cos \theta \tan \theta$$

Step 2: Use the Trigonometric Identity

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We can use the trigonometric identity $\tan \theta = \frac{\sin \theta}{\cos \theta}$ to rewrite the integrand as:

$$\cos \theta \tan \theta = \cos \theta \frac{\sin \theta}{\cos \theta} = \sin \theta$$

Step 3: Evaluate the Integral

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Now that we have simplified the integrand, we can evaluate the integral. The integral of $\sin \theta$ with respect to $\theta$ is:

$$\int \sin \theta \, d\theta = -\cos \theta + C$$

where $C$ is the constant of integration.

Step 4: Apply the Constant of Integration

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Since the integral is $\int \frac{1}{\sec \theta} \tan \theta \, d\theta$, we can apply the constant of integration $C$ to get:

$$\int \frac{1}{\sec \theta} \tan \theta \, d\theta = -\cos \theta + C$$

Conclusion

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In this article, we evaluated the given integral using various mathematical techniques and formulas. We simplified the integrand, used trigonometric identities, and evaluated the integral to arrive at the final answer. The integral is $\int \frac{1}{\sec \theta} \tan \theta \, d\theta = -\cos \theta + C$.

Final Answer

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The final answer is $\boxed{-\cos \theta + C}$.

Additional Information

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• The integral can be evaluated using various mathematical techniques and formulas.
• The trigonometric identity $\sec \theta = \frac{1}{\cos \theta}$ can be used to simplify the integrand.
• The trigonometric identity $\tan \theta = \frac{\sin \theta}{\cos \theta}$ can be used to rewrite the integrand.
• The integral of $\sin \theta$ with respect to $\theta$ is $-\cos \theta + C$.

Related Topics

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• Calculus
• Trigonometry
• Integration
• Differentiation

References

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• [1] Calculus by Michael Spivak
• [2] Trigonometry by I.M. Gelfand
• [3] Integration by David Guichard

Keywords

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• Integral
• Trigonometry
• Calculus
• Integration
• Differentiation
  

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Introduction

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In the previous article, we evaluated the integral $\int \frac{1}{\sec \theta} \tan \theta \, d\theta$. In this article, we will answer some frequently asked questions related to the integral.

Q&A

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Q: What is the integral of $\frac{1}{\sec \theta} \tan \theta$?

A: The integral of $\frac{1}{\sec \theta} \tan \theta$ is $-\cos \theta + C$.

Q: How do I simplify the integrand?

A: You can simplify the integrand by using the trigonometric identity $\sec \theta = \frac{1}{\cos \theta}$.

Q: What is the trigonometric identity for $\sec \theta$?

A: The trigonometric identity for $\sec \theta$ is $\sec \theta = \frac{1}{\cos \theta}$.

Q: How do I rewrite the integrand using the trigonometric identity for $\tan \theta$?

A: You can rewrite the integrand using the trigonometric identity for $\tan \theta$ as $\cos \theta \tan \theta = \cos \theta \frac{\sin \theta}{\cos \theta} = \sin \theta$.

Q: What is the integral of $\sin \theta$?

A: The integral of $\sin \theta$ is $-\cos \theta + C$.

Q: How do I apply the constant of integration?

A: You can apply the constant of integration $C$ to get the final answer.

Common Mistakes

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Mistake 1: Not simplifying the integrand

A: Make sure to simplify the integrand using the trigonometric identities.

Mistake 2: Not using the correct trigonometric identity

A: Use the correct trigonometric identity for $\sec \theta$ and $\tan \theta$.

Mistake 3: Not applying the constant of integration

A: Make sure to apply the constant of integration $C$ to get the final answer.

Tips and Tricks

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Tip 1: Use trigonometric identities to simplify the integrand

A: Use trigonometric identities to simplify the integrand and make it easier to evaluate.

Tip 2: Break down the integral into smaller parts

A: Break down the integral into smaller parts and evaluate each part separately.

Tip 3: Use the correct formula for the integral

A: Use the correct formula for the integral and make sure to apply the constant of integration.

Conclusion

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In this article, we answered some frequently asked questions related to the integral $\int \frac{1}{\sec \theta} \tan \theta \, d\theta$. We also discussed common mistakes and provided tips and tricks for evaluating the integral.

Final Answer

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The final answer is $\boxed{-\cos \theta + C}$.

Additional Information

---

• The integral can be evaluated using various mathematical techniques and formulas.
• The trigonometric identity $\sec \theta = \frac{1}{\cos \theta}$ can be used to simplify the integrand.
• The trigonometric identity $\tan \theta = \frac{\sin \theta}{\cos \theta}$ can be used to rewrite the integrand.
• The integral of $\sin \theta$ with respect to $\theta$ is $-\cos \theta + C$.

Related Topics

---

• Calculus
• Trigonometry
• Integration
• Differentiation

References

---

• [1] Calculus by Michael Spivak
• [2] Trigonometry by I.M. Gelfand
• [3] Integration by David Guichard

Keywords

---

• Integral
• Trigonometry
• Calculus
• Integration
• Differentiation