# Integrating tan^5(6x) sec^3(6x) - A Guide

• grothem
In summary, the purpose of integrating tan^5(6x) sec^3(6x) is to find the indefinite integral of the given trigonometric expression. To approach integrating tan^5(6x) sec^3(6x), you can use the substitution method and simplify the expression using the power rule and trigonometric identities. The general formula for integrating tan^n(x) sec^m(x) is derived from the power rule and the trigonometric identity cos^2(x) = 1 - tan^2(x). There are special cases when integrating tan^5(6x) sec^3(6x), which can be solved using the substitution method and trigonometric identities. To check if

## Homework Statement

$$\int tan^5(6x) sec^3(6x) dx$$

## The Attempt at a Solution

first off I set u=6x to get 1/6$$\int tan^5(u) sec^3(u) dx$$
then I used trig identities to put tangent in terms of secant and I came up with

$$\int sec^9(u)-3sec^7(u)+3sec^5(u)-sec^3(u) dx$$
Not sure where to go from here, or if I'm doing this the right way

$$\frac 1 6\int\tan^{5}u\sec^{3}udu$$

$$\frac 1 6\int\tan^{4}u\sec^{2}\sec u\tan udu$$

*$$\tan^{2}u+1=\sec^{2}u$$

Take it from here.

## 1. What is the purpose of integrating tan^5(6x) sec^3(6x)?

The purpose of integrating tan^5(6x) sec^3(6x) is to find the indefinite integral of the given trigonometric expression. This will help in solving problems involving motion, work, and other real-world applications.

## 2. How do I approach integrating tan^5(6x) sec^3(6x)?

To integrate tan^5(6x) sec^3(6x), you can use the substitution method by letting u = 6x and du = 6dx. Then, rewrite the expression using u, and use the power rule and the trigonometric identity cos^2(u) = 1 - tan^2(u) to simplify the expression.

## 3. What is the general formula for integrating tan^n(x) sec^m(x)?

The general formula for integrating tan^n(x) sec^m(x) is: ∫ tan^n(x) sec^m(x) dx = sec^m(x)/(m-1)tan^(n-1)(x) + (m-2)/(m-1) ∫ tan^(n-2)(x) sec^(m-2)(x) dx. This formula is derived from the power rule and the trigonometric identity cos^2(x) = 1 - tan^2(x).

## 4. Are there any special cases when integrating tan^5(6x) sec^3(6x)?

Yes, there are special cases when integrating tan^5(6x) sec^3(6x). If n or m is an odd number, the resulting integral will contain a secant function. In this case, you can use the substitution method to simplify the expression and then use trigonometric identities to solve the integral.

## 5. How do I check if my answer to the integral is correct?

You can check if your answer to the integral is correct by differentiating it. If the resulting expression is equal to the original integrand, then your answer is correct. You can also use online integration calculators or ask a colleague to verify your answer.

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