Remove A Factor Of $\sec^2 X$ From The Integrand:$\int \sec^6 X \tan^{10} X \, Dx = \int (\sec^4(x) \tan^{10}(x)) \sec^2 X \, Dx$---Rewrite In Terms Of $\tan X$:$\sec^4 X \tan^{10} X = (1 + \tan^2(x))^2

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Introduction

In calculus, we often encounter integrals that involve trigonometric functions, particularly the secant and tangent functions. When dealing with integrals of the form $\int \sec^m x \tan^n x \, dx$, we may need to remove a factor of $\sec^2 x$ from the integrand to simplify the integral. In this article, we will explore how to remove a factor of $\sec^2 x$ from the integrand and rewrite the integral in terms of $\tan x$.

Removing a Factor of $\sec^2 x$ from the Integrand

To remove a factor of $\sec^2 x$ from the integrand, we can use the following substitution:

$$\int \sec^6 x \tan^{10} x \, dx = \int (\sec^4(x) \tan^{10}(x)) \sec^2 x \, dx$$

This substitution allows us to rewrite the integral as a product of two integrals, one of which is a power of $\sec x$ and the other is a power of $\tan x$. We can then use the following identity to simplify the integral:

$$\sec^4 x \tan^{10} x = (1 + \tan^2(x))^2$$

This identity allows us to rewrite the integral in terms of $\tan x$.

Rewriting in Terms of $\tan x$

To rewrite the integral in terms of $\tan x$, we can use the following substitution:

$$u = \tan x$$

This substitution allows us to rewrite the integral as:

$$\int (1 + u^2)^2 u^{10} \, du$$

We can then use the following identity to simplify the integral:

$$(1 + u^2)^2 = 1 + 2u^2 + u^4$$

This identity allows us to rewrite the integral as:

$$\int (1 + 2u^2 + u^4) u^{10} \, du$$

We can then use the following power rule of integration to simplify the integral:

$$\int u^n \, du = \frac{u^{n+1}}{n+1} + C$$

This power rule allows us to rewrite the integral as:

$$\int (1 + 2u^2 + u^4) u^{10} \, du = \frac{u^{11}}{11} + \frac{2u^{13}}{13} + \frac{u^{15}}{15} + C$$

We can then substitute back $u = \tan x$ to obtain the final result:

$$\int \sec^6 x \tan^{10} x \, dx = \frac{\tan^{11} x}{11} + \frac{2\tan^{13} x}{13} + \frac{\tan^{15} x}{15} + C$$

Conclusion

In this article, we have shown how to remove a factor of $\sec^2 x$ from the integrand and rewrite the integral in terms of $\tan x$. We have used the following substitution to simplify the integral:

$$\int \sec^6 x \tan^{10} x \, dx = \int (\sec^4(x) \tan^{10}(x)) \sec^2 x \, dx$$

We have then used the following identity to rewrite the integral in terms of $\tan x$:

$$\sec^4 x \tan^{10} x = (1 + \tan^2(x))^2$$

We have also used the following power rule of integration to simplify the integral:

$$\int u^n \, du = \frac{u^{n+1}}{n+1} + C$$

This power rule has allowed us to rewrite the integral as:

$$\int (1 + 2u^2 + u^4) u^{10} \, du = \frac{u^{11}}{11} + \frac{2u^{13}}{13} + \frac{u^{15}}{15} + C$$

We have then substituted back $u = \tan x$ to obtain the final result:

$$\int \sec^6 x \tan^{10} x \, dx = \frac{\tan^{11} x}{11} + \frac{2\tan^{13} x}{13} + \frac{\tan^{15} x}{15} + C$$

References

• [1] Calculus, 3rd edition, Michael Spivak
• [2] Calculus, 2nd edition, James Stewart
• [3] Trigonometry, 2nd edition, Charles P. McKeague

Discussion

This article has shown how to remove a factor of $\sec^2 x$ from the integrand and rewrite the integral in terms of $\tan x$. The substitution used in this article is a common technique used in calculus to simplify integrals. The power rule of integration used in this article is also a common technique used in calculus to simplify integrals.

The final result obtained in this article is a common result in calculus, and it can be used to solve a variety of problems involving integrals of trigonometric functions.

Related Articles

• [1] Integration by Substitution: This article discusses how to use substitution to simplify integrals.
• [2] Integration by Parts: This article discusses how to use integration by parts to simplify integrals.
• [3] Trigonometric Integrals: This article discusses how to simplify integrals involving trigonometric functions.

Comments

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Introduction

In our previous article, we discussed how to remove a factor of $\sec^2 x$ from the integrand and rewrite the integral in terms of $\tan x$. In this article, we will answer some common questions that readers may have about this topic.

Q: What is the purpose of removing a factor of $\sec^2 x$ from the integrand?

A: The purpose of removing a factor of $\sec^2 x$ from the integrand is to simplify the integral and make it easier to evaluate. By removing this factor, we can rewrite the integral in terms of $\tan x$, which can be easier to work with.

Q: How do I know when to remove a factor of $\sec^2 x$ from the integrand?

A: You should remove a factor of $\sec^2 x$ from the integrand when you see a power of $\sec x$ multiplied by a power of $\tan x$. This is because the identity $\sec^4 x \tan^{10} x = (1 + \tan^2(x))^2$ allows us to rewrite the integral in terms of $\tan x$.

Q: What is the substitution used to remove a factor of $\sec^2 x$ from the integrand?

A: The substitution used to remove a factor of $\sec^2 x$ from the integrand is:

$$\int \sec^6 x \tan^{10} x \, dx = \int (\sec^4(x) \tan^{10}(x)) \sec^2 x \, dx$$

This substitution allows us to rewrite the integral as a product of two integrals, one of which is a power of $\sec x$ and the other is a power of $\tan x$.

Q: How do I evaluate the integral after removing a factor of $\sec^2 x$ from the integrand?

A: After removing a factor of $\sec^2 x$ from the integrand, you can use the power rule of integration to evaluate the integral. The power rule of integration states that:

$$\int u^n \, du = \frac{u^{n+1}}{n+1} + C$$

You can use this rule to evaluate the integral after removing a factor of $\sec^2 x$ from the integrand.

Q: What are some common mistakes to avoid when removing a factor of $\sec^2 x$ from the integrand?

A: Some common mistakes to avoid when removing a factor of $\sec^2 x$ from the integrand include:

• Not recognizing the identity $\sec^4 x \tan^{10} x = (1 + \tan^2(x))^2$ and not using it to rewrite the integral in terms of $\tan x$.
• Not using the power rule of integration to evaluate the integral after removing a factor of $\sec^2 x$ from the integrand.
• Not checking the work and making sure that the integral is evaluated correctly.

Q: How can I practice removing a factor of $\sec^2 x$ from the integrand?

A: You can practice removing a factor of $\sec^2 x$ from the integrand by working through examples and exercises. You can also try removing a factor of $\sec^2 x$ from the integrand in different types of integrals, such as integrals of the form $\int \sec^m x \tan^n x \, dx$.

Conclusion

In this article, we have answered some common questions that readers may have about removing a factor of $\sec^2 x$ from the integrand and rewriting the integral in terms of $\tan x$. We hope that this article has been helpful in clarifying any confusion and providing additional practice and examples.

References

• [1] Calculus, 3rd edition, Michael Spivak
• [2] Calculus, 2nd edition, James Stewart
• [3] Trigonometry, 2nd edition, Charles P. McKeague

Discussion

This article has provided additional practice and examples for removing a factor of $\sec^2 x$ from the integrand and rewriting the integral in terms of $\tan x$. The questions and answers in this article have covered a range of topics, from the purpose of removing a factor of $\sec^2 x$ from the integrand to common mistakes to avoid.

Related Articles

• [1] Integration by Substitution: This article discusses how to use substitution to simplify integrals.
• [2] Integration by Parts: This article discusses how to use integration by parts to simplify integrals.
• [3] Trigonometric Integrals: This article discusses how to simplify integrals involving trigonometric functions.

Comments

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