Evaluate The Integral: ${ \int \frac{1-\cos X}{1+\cos X} , D X }$

Introduction

In this article, we will delve into the world of calculus and evaluate the integral of a trigonometric function. The given integral is ${ \int \frac{1-\cos x}{1+\cos x} , d x }$. This type of integral is commonly encountered in mathematics and physics, and its evaluation requires a deep understanding of trigonometric identities and integration techniques.

Trigonometric Identities

Before we proceed with the evaluation of the integral, let's recall some essential trigonometric identities that will be useful in this process.

• Pythagorean Identity: $\sin^2 x + \cos^2 x = 1$
• Double Angle Formula: $\sin 2x = 2\sin x \cos x$
• Half Angle Formula: $\cos \frac{x}{2} = \pm \sqrt{\frac{1 + \cos x}{2}}$

These identities will help us simplify the given integral and make it more manageable.

Simplifying the Integral

Let's start by simplifying the given integral using trigonometric identities.

{ \int \frac{1-\cos x}{1+\cos x} \, d x \}

We can rewrite the numerator as:

{ 1 - \cos x = \sin^2 \frac{x}{2} \}

Using the half angle formula, we can rewrite the numerator as:

{ \sin^2 \frac{x}{2} = \frac{1 - \cos x}{2} \}

Now, let's substitute this expression into the original integral:

{ \int \frac{\frac{1 - \cos x}{2}}{1 + \cos x} \, d x \}

Simplifying the expression, we get:

{ \int \frac{1 - \cos x}{2(1 + \cos x)} \, d x \}

We can further simplify the expression by multiplying the numerator and denominator by 2:

{ \int \frac{1 - \cos x}{2(1 + \cos x)} \cdot \frac{2}{2} \, d x \}

This simplifies to:

{ \int \frac{1 - \cos x}{1 + \cos x} \, d x \}

Evaluating the Integral

Now that we have simplified the integral, let's proceed with its evaluation.

We can start by making a substitution:

{ u = \tan \frac{x}{2} \}

Differentiating both sides with respect to x, we get:

{ \frac{du}{dx} = \frac{1}{2} \sec^2 \frac{x}{2} \}

We can rewrite this expression as:

{ \frac{du}{dx} = \frac{1}{2} (1 + \tan^2 \frac{x}{2}) \}

Using the Pythagorean identity, we can rewrite this expression as:

{ \frac{du}{dx} = \frac{1}{2} (1 + u^2) \}

Now, let's substitute this expression into the original integral:

{ \int \frac{1 - \cos x}{1 + \cos x} \, d x \}

Using the substitution $u = \tan \frac{x}{2}$, we get:

{ \int \frac{1 - \cos x}{1 + \cos x} \, d x = \int \frac{1 - \frac{1 - u^2}{1 + u^2}}{1 + \frac{1 - u^2}{1 + u^2}} \, du \}

Simplifying the expression, we get:

{ \int \frac{1 - \cos x}{1 + \cos x} \, d x = \int \frac{2u}{1 + u^2} \, du \}

This is a standard integral that can be evaluated using the substitution $v = 1 + u^2$.

Substitution Method

Let's use the substitution method to evaluate the integral.

{ v = 1 + u^2 \}

Differentiating both sides with respect to u, we get:

{ \frac{dv}{du} = 2u \}

We can rewrite this expression as:

{ \frac{dv}{du} = 2u \}

Now, let's substitute this expression into the original integral:

{ \int \frac{2u}{1 + u^2} \, du \}

Using the substitution $v = 1 + u^2$, we get:

{ \int \frac{2u}{1 + u^2} \, du = \int \frac{1}{v} \, dv \}

This is a standard integral that can be evaluated using the power rule of integration.

Power Rule of Integration

The power rule of integration states that:

{ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \}

Using this rule, we can evaluate the integral:

{ \int \frac{1}{v} \, dv = \ln |v| + C \}

Substituting back $v = 1 + u^2$, we get:

{ \int \frac{1}{v} \, dv = \ln |1 + u^2| + C \}

Back Substitution

Now that we have evaluated the integral, let's substitute back the original variable x.

{ u = \tan \frac{x}{2} \}

Using the trigonometric identity $\tan^2 \frac{x}{2} + 1 = \sec^2 \frac{x}{2}$, we can rewrite the expression as:

{ \ln |1 + u^2| = \ln |\sec^2 \frac{x}{2}| \}

Using the logarithmic identity $\ln |a^b| = b \ln |a|$, we can rewrite the expression as:

{ \ln |\sec^2 \frac{x}{2}| = 2 \ln |\sec \frac{x}{2}| \}

Final Answer

The final answer to the integral is:

{ \int \frac{1-\cos x}{1+\cos x} \, d x = 2 \ln |\sec \frac{x}{2}| + C \}

This is the final answer to the integral. We have used various trigonometric identities and integration techniques to simplify and evaluate the integral.

Conclusion

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Introduction

In our previous article, we evaluated the integral of a trigonometric function using various techniques such as substitution, power rule of integration, and logarithmic identities. In this article, we will provide a Q&A section to help clarify any doubts or questions that readers may have.

Q: What is the main concept behind evaluating the integral of a trigonometric function?

A: The main concept behind evaluating the integral of a trigonometric function is to use various techniques such as substitution, power rule of integration, and logarithmic identities to simplify the integral and make it more manageable.

Q: What are some common trigonometric identities that are used in evaluating the integral of a trigonometric function?

A: Some common trigonometric identities that are used in evaluating the integral of a trigonometric function include:

• Pythagorean Identity: $\sin^2 x + \cos^2 x = 1$
• Double Angle Formula: $\sin 2x = 2\sin x \cos x$
• Half Angle Formula: $\cos \frac{x}{2} = \pm \sqrt{\frac{1 + \cos x}{2}}$

Q: What is the substitution method used in evaluating the integral of a trigonometric function?

A: The substitution method is used in evaluating the integral of a trigonometric function by substituting a new variable into the integral. This new variable is often a trigonometric function such as $\tan \frac{x}{2}$ or $\sin \frac{x}{2}$.

Q: What is the power rule of integration used in evaluating the integral of a trigonometric function?

A: The power rule of integration is used in evaluating the integral of a trigonometric function by integrating the function with respect to the new variable. This rule states that:

{ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \}

Q: What is the logarithmic identity used in evaluating the integral of a trigonometric function?

A: The logarithmic identity used in evaluating the integral of a trigonometric function is:

{ \ln |a^b| = b \ln |a| \}

Q: What is the final answer to the integral of a trigonometric function?

A: The final answer to the integral of a trigonometric function is:

{ \int \frac{1-\cos x}{1+\cos x} \, d x = 2 \ln |\sec \frac{x}{2}| + C \}

Conclusion

In this Q&A article, we have provided answers to some common questions that readers may have about evaluating the integral of a trigonometric function. We have also provided a brief overview of the main concepts and techniques used in evaluating the integral of a trigonometric function. We hope that this article has been helpful in clarifying any doubts or questions that readers may have.

Frequently Asked Questions

• Q: What is the main concept behind evaluating the integral of a trigonometric function? A: The main concept behind evaluating the integral of a trigonometric function is to use various techniques such as substitution, power rule of integration, and logarithmic identities to simplify the integral and make it more manageable.
• Q: What are some common trigonometric identities that are used in evaluating the integral of a trigonometric function? A: Some common trigonometric identities that are used in evaluating the integral of a trigonometric function include the Pythagorean identity, double angle formula, and half angle formula.
• Q: What is the substitution method used in evaluating the integral of a trigonometric function? A: The substitution method is used in evaluating the integral of a trigonometric function by substituting a new variable into the integral.
• Q: What is the power rule of integration used in evaluating the integral of a trigonometric function? A: The power rule of integration is used in evaluating the integral of a trigonometric function by integrating the function with respect to the new variable.
• Q: What is the logarithmic identity used in evaluating the integral of a trigonometric function? A: The logarithmic identity used in evaluating the integral of a trigonometric function is $\ln |a^b| = b \ln |a|$.
• Q: What is the final answer to the integral of a trigonometric function? A: The final answer to the integral of a trigonometric function is $2 \ln |\sec \frac{x}{2}| + C$.

Additional Resources

• Calculus Textbook: A calculus textbook that provides a comprehensive overview of calculus, including integration and trigonometric functions.
• Online Calculus Resources: Online resources that provide additional information and practice problems for calculus, including integration and trigonometric functions.
• Calculus Software: Software that provides a graphical interface for calculus, including integration and trigonometric functions.

Conclusion

In this article, we have provided a Q&A section to help clarify any doubts or questions that readers may have about evaluating the integral of a trigonometric function. We have also provided a brief overview of the main concepts and techniques used in evaluating the integral of a trigonometric function. We hope that this article has been helpful in clarifying any doubts or questions that readers may have.