The given integral is:

$\int 10 \tan^3(x) \, dx$

Let's evaluate this step-by-step.

Step 1: Extract the constant

The integral has a constant factor of 10, which can be taken outside the integral: $10 \int \tan^3(x) \, dx$

Step 2: Express $\tan^3(x)$ in terms of $\tan(x)$ and $\sec^2(x)$

We can rewrite $\tan^3(x)$ as: $\tan^3(x) = \tan^2(x) \cdot \tan(x)$ Recall that $\tan^2(x) = \sec^2(x) - 1$. Substituting this, we have: $\tan^3(x) = (\sec^2(x) - 1) \tan(x)$

Thus, the integral becomes: $10 \int (\sec^2(x) - 1) \tan(x) \, dx$

Step 3: Separate the integral

We can split this into two integrals: $10 \left( \int \sec^2(x) \tan(x) \, dx - \int \tan(x) \, dx \right)$

Step 4: Evaluate each integral

1. For $\int \sec^2(x) \tan(x) \, dx$: Let $u = \tan(x)$, then $du = \sec^2(x) \, dx$. Therefore, $\int \sec^2(x) \tan(x) \, dx = \int u \, du = \frac{u^2}{2} = \frac{\tan^2(x)}{2}$

2. For $\int \tan(x) \, dx$: Recall the standard result: $\int \tan(x) \, dx = -\ln |\cos(x)|$

Step 5: Combine the results

Substituting back, we get: $10 \left( \frac{\tan^2(x)}{2} + \ln |\cos(x)| \right) + C$ where $C$ is the constant of integration.

Final Answer

$5 \tan^2(x) + 10 \ln |\cos(x)| + C$

Would you like more details or have any questions about this solution?

Related Questions

1. How do you evaluate the integral of $\tan^n(x)$ for any positive integer $n$?
2. What are the integrals of other trigonometric functions like $\sec^3(x)$ or $\csc^3(x)$?
3. How can you verify an integral result using differentiation?
4. What substitution methods are commonly useful in integrating trigonometric expressions?
5. How would you handle the integral if it had different limits?

Tip: Remember that trigonometric identities, such as $\tan^2(x) = \sec^2(x) - 1$, are often helpful for simplifying integrals involving powers of trigonometric functions.