Rewrite By Removing A Factor Of $\tan^2(x$\] From The Integrand:$\int (\tan(x))^6 \, Dx = \int \tan^4(x) \, \tan^2(x) \, Dx$Rewrite Using The Pythagorean Identity $1 + \tan^2(x) = \sec^2(x$\]:$\int (\tan(x))^4 \tan^2(x) \,

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Introduction

When dealing with trigonometric integrals, it's often helpful to simplify the integrand by removing a factor of $\tan^2(x)$ from the integrand. This can be achieved by using the Pythagorean identity $1 + \tan^2(x) = \sec^2(x)$, which allows us to rewrite the integrand in a more manageable form. In this article, we will explore how to rewrite the integral $\int (\tan(x))^6 \, dx$ by removing a factor of $\tan^2(x)$ from the integrand.

Rewriting the Integrand

To rewrite the integrand, we can start by expressing $\tan^6(x)$ as $\tan^4(x) \, \tan^2(x)$. This allows us to rewrite the integral as:

$$\int (\tan(x))^6 \, dx = \int \tan^4(x) \, \tan^2(x) \, dx$$

Using the Pythagorean Identity

Now, we can use the Pythagorean identity $1 + \tan^2(x) = \sec^2(x)$ to rewrite the integrand. By rearranging the identity, we can express $\tan^2(x)$ as $\sec^2(x) - 1$. Substituting this expression into the integrand, we get:

$$\int \tan^4(x) \, \tan^2(x) \, dx = \int \tan^4(x) \, (\sec^2(x) - 1) \, dx$$

Simplifying the Integrand

To simplify the integrand, we can expand the expression $(\sec^2(x) - 1)$ to get:

$$\int \tan^4(x) \, (\sec^2(x) - 1) \, dx = \int \tan^4(x) \, \sec^2(x) \, dx - \int \tan^4(x) \, dx$$

Evaluating the Integrals

Now, we can evaluate the two integrals separately. The first integral can be evaluated using the substitution $u = \tan(x)$, which gives:

$$\int \tan^4(x) \, \sec^2(x) \, dx = \int u^4 \, du = \frac{u^5}{5} + C$$

Substituting back $u = \tan(x)$, we get:

$$\int \tan^4(x) \, \sec^2(x) \, dx = \frac{\tan^5(x)}{5} + C$$

The second integral can be evaluated using the substitution $u = \tan^2(x)$, which gives:

$$\int \tan^4(x) \, dx = \int u^2 \, du = \frac{u^3}{3} + C$$

Substituting back $u = \tan^2(x)$, we get:

$$\int \tan^4(x) \, dx = \frac{\tan^6(x)}{3} + C$$

Combining the Results

Now, we can combine the results of the two integrals to get:

$$\int (\tan(x))^6 \, dx = \frac{\tan^5(x)}{5} + \frac{\tan^6(x)}{3} + C$$

Conclusion

In this article, we have shown how to rewrite the integral $\int (\tan(x))^6 \, dx$ by removing a factor of $\tan^2(x)$ from the integrand. We used the Pythagorean identity $1 + \tan^2(x) = \sec^2(x)$ to rewrite the integrand, and then evaluated the resulting integrals using substitution. The final result is a simplified expression for the integral, which can be used to solve a variety of problems in trigonometry.

Discussion

The Pythagorean identity $1 + \tan^2(x) = \sec^2(x)$ is a fundamental concept in trigonometry, and is used extensively in the evaluation of trigonometric integrals. By using this identity, we can rewrite the integrand in a more manageable form, and then evaluate the resulting integrals using substitution. This technique is useful in a variety of applications, including the evaluation of trigonometric integrals and the solution of trigonometric equations.

Applications

The technique of rewriting the integrand by removing a factor of $\tan^2(x)$ from the integrand has a variety of applications in trigonometry. For example, it can be used to evaluate the integral $\int (\tan(x))^6 \, dx$, which is a common integral in trigonometry. It can also be used to solve trigonometric equations, such as the equation $\tan(x) = 2$. By using the Pythagorean identity, we can rewrite the equation in a more manageable form, and then solve for $x$.

Future Work

In future work, we plan to explore other techniques for rewriting the integrand, such as using the identity $\tan(x) = \frac{\sin(x)}{\cos(x)}$. We also plan to investigate the use of substitution in the evaluation of trigonometric integrals, and to explore the application of these techniques to a variety of problems in trigonometry.

References

• [1] "Trigonometry" by Michael Corral
• [2] "Calculus" by Michael Spivak
• [3] "Trigonometric Integrals" by Wolfram MathWorld

Glossary

• Pythagorean identity: The identity $1 + \tan^2(x) = \sec^2(x)$, which is used extensively in the evaluation of trigonometric integrals.
• Substitution: A technique used to evaluate integrals by substituting a new variable for the original variable.
• Trigonometric integral: An integral that involves trigonometric functions, such as $\sin(x)$ and $\cos(x)$.
• Trigonometric equation: An equation that involves trigonometric functions, such as $\tan(x) = 2$.
  

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Introduction

In our previous article, we showed how to rewrite the integral $\int (\tan(x))^6 \, dx$ by removing a factor of $\tan^2(x)$ from the integrand. We used the Pythagorean identity $1 + \tan^2(x) = \sec^2(x)$ to rewrite the integrand, and then evaluated the resulting integrals using substitution. In this article, we will answer some common questions about rewriting the integrand by removing a factor of $\tan^2(x)$.

Q: What is the Pythagorean identity?

A: The Pythagorean identity is the equation $1 + \tan^2(x) = \sec^2(x)$. This identity is used extensively in the evaluation of trigonometric integrals.

Q: How do I use the Pythagorean identity to rewrite the integrand?

A: To use the Pythagorean identity to rewrite the integrand, you can substitute $\sec^2(x) - 1$ for $\tan^2(x)$ in the integrand. This will give you a new expression for the integrand that can be evaluated using substitution.

Q: What is substitution?

A: Substitution is a technique used to evaluate integrals by substituting a new variable for the original variable. In the case of the Pythagorean identity, we substitute $u = \tan(x)$ for the original variable $x$.

Q: How do I evaluate the integral after using substitution?

A: To evaluate the integral after using substitution, you can integrate the new expression with respect to the new variable. In the case of the Pythagorean identity, we integrate $u^4 \, du$ with respect to $u$.

Q: What are some common applications of rewriting the integrand by removing a factor of $\tan^2(x)$?

A: Some common applications of rewriting the integrand by removing a factor of $\tan^2(x)$ include evaluating the integral $\int (\tan(x))^6 \, dx$, solving trigonometric equations, and evaluating trigonometric integrals.

Q: Can I use the Pythagorean identity to rewrite the integrand for any trigonometric function?

A: No, the Pythagorean identity can only be used to rewrite the integrand for trigonometric functions that involve $\tan(x)$. However, there are other identities that can be used to rewrite the integrand for other trigonometric functions.

Q: What are some common mistakes to avoid when rewriting the integrand by removing a factor of $\tan^2(x)$?

A: Some common mistakes to avoid when rewriting the integrand by removing a factor of $\tan^2(x)$ include:

• Not using the Pythagorean identity correctly
• Not substituting the new variable correctly
• Not integrating the new expression correctly
• Not checking the final answer for errors

Q: How can I practice rewriting the integrand by removing a factor of $\tan^2(x)$?

A: You can practice rewriting the integrand by removing a factor of $\tan^2(x)$ by working through examples and exercises in a textbook or online resource. You can also try rewriting the integrand for different trigonometric functions and checking your answers for errors.

Conclusion

In this article, we have answered some common questions about rewriting the integrand by removing a factor of $\tan^2(x)$. We have discussed the Pythagorean identity, substitution, and common applications of rewriting the integrand. We have also provided some tips for avoiding common mistakes and practicing rewriting the integrand.

Glossary

• Pythagorean identity: The identity $1 + \tan^2(x) = \sec^2(x)$, which is used extensively in the evaluation of trigonometric integrals.
• Substitution: A technique used to evaluate integrals by substituting a new variable for the original variable.
• Trigonometric integral: An integral that involves trigonometric functions, such as $\sin(x)$ and $\cos(x)$.
• Trigonometric equation: An equation that involves trigonometric functions, such as $\tan(x) = 2$.

References

• [1] "Trigonometry" by Michael Corral
• [2] "Calculus" by Michael Spivak
• [3] "Trigonometric Integrals" by Wolfram MathWorld

Further Reading

• "Trigonometric Identities" by Wolfram MathWorld
• "Substitution Method" by Math Open Reference
• "Trigonometric Integrals" by Khan Academy