 Evaluate the integral. (Remember the constant of integration.)
	4 tan(x) sec3(x) dx


To evaluate the integral \( \int 4 \tan(x) \sec^3(x) \, dx \), we can proceed by using a substitution method.

### Step 1: Use substitution
Let’s use the substitution \( u = \sec(x) \), which gives:

\[
du = \sec(x) \tan(x) \, dx
\]

We can now rewrite the integral in terms of \( u \). First, recognize that \( \sec^3(x) = u^3 \), and \( \sec(x) \tan(x) \, dx = du \), so the integral becomes:

\[
\int 4 \tan(x) \sec^3(x) \, dx = \int 4 u^2 \, du
\]

### Step 2: Integrate
Now integrate \( 4 u^2 \):

\[
\int 4 u^2 \, du = \frac{4 u^3}{3} + C
\]

### Step 3: Substitute back \( u = \sec(x) \)
Now, substitute back \( u = \sec(x) \):

\[
\frac{4 \sec^3(x)}{3} + C
\]

### Final Answer:
Thus, the integral is:

\[
\frac{4 \sec^3(x)}{3} + C
\]

Would you like more details or have any questions?

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Here are 5 related questions to further explore this topic:
1. What are the other common techniques to solve integrals involving trigonometric functions?
2. How would the solution change if the integral were \( \int \tan(x) \sec^2(x) \, dx \)?
3. What is the geometric interpretation of \( \sec(x) \) in relation to the unit circle?
4. How would you approach an integral involving \( \tan^2(x) \)?
5. Can we use integration by parts to solve this type of trigonometric integral?

**Tip:** In integrals involving powers of secant and tangent, substitution with \( u = \sec(x) \) or \( u = \tan(x) \) can often simplify the expression.

Evaluate the integral. (Remember the constant of integration.) 4 tan(x) sec^3(x) dx

This page covers the step-by-step solution to the integral ∫ 4 tan(x) sec^3(x) dx using the substitution method. Learn how to use trigonometric identities and substitution to solve integrals involving secant and tangent functions. Suitable for advanced high school or early college students.

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