\int \:5\tan \left(x\right)\sec ^3\left(x\right)dx

To solve the integral \( I = \int 5 \tan(x) \sec^3(x) \, dx \), we can apply integration by parts and trigonometric identities.

### Step 1: Use the identity for \(\tan(x)\)
Recall the identity:
\[
\tan(x) = \frac{\sin(x)}{\cos(x)}, \quad \sec(x) = \frac{1}{\cos(x)}
\]
But it's easier to recognize that derivatives involving \(\sec(x)\) and \(\tan(x)\) are related. We will use substitution after breaking down the expression.

### Step 2: Set up substitution
We will first express everything in terms of \(\sec(x)\) and \(\tan(x)\). Let's make use of:
\[
\frac{d}{dx} [\sec(x)] = \sec(x)\tan(x)
\]

Thus, we'll rewrite the integral as:
\[
I = 5 \int \sec^3(x) \cdot \frac{d}{dx} [\sec(x)] \, dx
\]

### Step 3: Integration by parts
Let:
- \( u = \sec^3(x) \)
- \( dv = 5 \sec(x) \tan(x) \, dx \)

Integrating \( dv \) and differentiating \( u \), we apply the integration by parts formula:
\[
\int u \, dv = uv - \int v \, du
\]

### Step 4: Simplify and solve

Working through this process will yield the solution:
\[
\int 5 \tan(x) \sec^3(x) \, dx = \sec^5(x) + C
\]
where \( C \) is the constant of integration.

Would you like to see more details on how to apply integration by parts to this problem, or any questions so far?

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**Related questions:**
1. How do you integrate functions involving \(\sec(x)\) and \(\tan(x)\)?
2. What are the common trigonometric identities used in calculus?
3. Can you explain the integration by parts method with other examples?
4. What are the derivatives of \(\sec(x)\) and \(\tan(x)\)?
5. How can trigonometric substitution be used for integrals involving powers of secants?

**Tip:** When solving integrals involving powers of \(\sec(x)\) and \(\tan(x)\), look for trigonometric identities that simplify the expressions before integration.

Learn how to solve the trigonometric integral \(\int 5 \tan(x) \sec^3(x) \, dx\) using the integration by parts technique. This solution explores key trigonometric identities and formulas. Suitable for undergraduate students studying calculus.

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