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

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Introduction

In calculus, we often encounter trigonometric integrals that involve powers of secant and tangent functions. One common technique for simplifying these integrals is to remove a factor of $\sec^2 x$ from the integrand, which can then be rewritten in terms of $\tan x$. In this article, we will explore how to remove a factor of $\sec^2 x$ from the integrand and rewrite the resulting expression 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 trigonometric identity:

$$\sec^2 x = 1 + \tan^2 x$$

Using this identity, we can rewrite the integrand as follows:

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

Now, we can substitute the expression for $\sec^2 x$ in terms of $\tan x$ into the integrand:

$$\int \left(\sec^4(x) \tan^{10}(x)\right) \sec^2 x \, dx = \int \left(\sec^4(x) \tan^{10}(x)\right) (1 + \tan^2 x) \, dx$$

Rewriting in Terms of $\tan x$

To rewrite the integrand in terms of $\tan x$, we can use the following trigonometric identity:

$$\sec^4 x = \frac{1}{\cos^4 x} = \frac{1}{(1 - \tan^2 x)^2}$$

Using this identity, we can rewrite the integrand as follows:

$$\int \left(\sec^4(x) \tan^{10}(x)\right) (1 + \tan^2 x) \, dx = \int \left(\frac{1}{(1 - \tan^2 x)^2} \tan^{10}(x)\right) (1 + \tan^2 x) \, dx$$

Now, we can simplify the integrand by combining the terms:

$$\int \left(\frac{1}{(1 - \tan^2 x)^2} \tan^{10}(x)\right) (1 + \tan^2 x) \, dx = \int \frac{\tan^{12} x}{(1 - \tan^2 x)^2} \, dx$$

Simplifying the Integrand

To simplify the integrand, we can use the following trigonometric identity:

$$\frac{1}{(1 - \tan^2 x)^2} = \frac{1}{\cos^4 x} = \frac{1}{(1 - \tan^2 x)^2} = \frac{1}{(1 - \tan^2 x)^2} \cdot \frac{(1 + \tan^2 x)^2}{(1 + \tan^2 x)^2}$$

Using this identity, we can rewrite the integrand as follows:

$$\int \frac{\tan^{12} x}{(1 - \tan^2 x)^2} \, dx = \int \frac{\tan^{12} x (1 + \tan^2 x)^2}{(1 - \tan^2 x)^2 (1 + \tan^2 x)^2} \, dx$$

Now, we can simplify the integrand by canceling out the common factors:

$$\int \frac{\tan^{12} x (1 + \tan^2 x)^2}{(1 - \tan^2 x)^2 (1 + \tan^2 x)^2} \, dx = \int \frac{\tan^{12} x}{(1 - \tan^2 x)^2} \, dx$$

Conclusion

In this article, we have shown how to remove a factor of $\sec^2 x$ from the integrand and rewrite the resulting expression in terms of $\tan x$. We have used trigonometric identities to simplify the integrand and rewrite it in a more manageable form. This technique can be useful for simplifying trigonometric integrals and making them easier to evaluate.

Final Answer

The final answer is not a numerical value, but rather a simplified expression for the integrand:

$$\int \frac{\tan^{12} x}{(1 - \tan^2 x)^2} \, dx$$

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Introduction

In our previous article, we explored how to remove a factor of $\sec^2 x$ from the integrand and rewrite the resulting expression in terms of $\tan x$. In this article, we will answer some common questions related to this topic and provide additional insights and examples.

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 expression and make it easier to evaluate. By rewriting the integrand in terms of $\tan x$, we can often use trigonometric identities and formulas to simplify the expression and make it more manageable.

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 whenever you see it in the expression. This is because $\sec^2 x$ is a fundamental trigonometric identity that can be used to rewrite the expression in terms of $\tan x$.

Q: What are some common trigonometric identities that I can use to simplify the integrand?

A: Some common trigonometric identities that you can use to simplify the integrand include:

• $\sec^2 x = 1 + \tan^2 x$
• $\tan^2 x = \sec^2 x - 1$
• $\sec^4 x = \frac{1}{\cos^4 x} = \frac{1}{(1 - \tan^2 x)^2}$
• $\tan^2 x = \frac{\sin^2 x}{\cos^2 x}$

Q: How do I rewrite the integrand in terms of $\tan x$?

A: To rewrite the integrand in terms of $\tan x$, you can use the following steps:

1. Identify the factor of $\sec^2 x$ in the integrand.
2. Use the trigonometric identity $\sec^2 x = 1 + \tan^2 x$ to rewrite the factor in terms of $\tan x$.
3. Simplify the expression by combining the terms.
4. Use additional trigonometric identities to simplify the expression further.

Q: What are some examples of integrals that can be simplified using this technique?

A: Some examples of integrals that can be simplified using this technique include:

• $\int \sec^6 x \tan^{10} x \, dx$
• $\int \sec^4 x \tan^8 x \, dx$
• $\int \sec^2 x \tan^6 x \, dx$

Q: Can I use this technique to simplify other types of integrals?

A: Yes, you can use this technique to simplify other types of integrals that involve trigonometric functions. However, you may need to use different trigonometric identities and formulas to simplify the expression.

Conclusion

In this article, we have answered some common questions related to removing a factor of $\sec^2 x$ from the integrand and rewriting the resulting expression in terms of $\tan x$. We have also provided additional insights and examples to help you understand this technique and apply it to different types of integrals.

Final Answer

The final answer is not a numerical value, but rather a simplified expression for the integrand. By using the technique of removing a factor of $\sec^2 x$ from the integrand and rewriting the resulting expression in terms of $\tan x$, you can simplify complex trigonometric integrals and make them easier to evaluate.