To solve the integral $\int_{0}^{\frac{\pi}{4}} 4 \tan(x) \sec^2(x) \, dx$ using the Fundamental Theorem of Calculus (Part 1 or Part 2), follow these steps:

Step 1: Analyze the integral

The integrand is $4 \tan(x) \sec^2(x)$. Notice that $\sec^2(x)$ is the derivative of $\tan(x)$, which suggests substitution.

Step 2: Substitution

Let: $u = \tan(x) \implies \frac{du}{dx} = \sec^2(x) \quad \text{or} \quad du = \sec^2(x) \, dx.$

The limits of integration also change:

• When $x = 0$, $u = \tan(0) = 0$.
• When $x = \frac{\pi}{4}$, $u = \tan\left(\frac{\pi}{4}\right) = 1$.

Substitute into the integral: $\int_{0}^{\frac{\pi}{4}} 4 \tan(x) \sec^2(x) \, dx = \int_{0}^{1} 4u \, du.$

Step 3: Solve the integral

The integral simplifies to: $\int_{0}^{1} 4u \, du = 4 \int_{0}^{1} u \, du.$ Using the power rule for integration: $\int u \, du = \frac{u^2}{2}.$

So: $4 \int_{0}^{1} u \, du = 4 \left[\frac{u^2}{2}\right]_{0}^{1} = 4 \cdot \left(\frac{1^2}{2} - \frac{0^2}{2}\right).$

Simplify: $4 \cdot \frac{1}{2} = 2.$

Final Answer:

$\int_{0}^{\frac{\pi}{4}} 4 \tan(x) \sec^2(x) \, dx = 2.$

Would you like me to explain any specific step in more detail? Here are some related questions for further exploration:

1. How does the substitution method simplify solving integrals?
2. What is the role of the Fundamental Theorem of Calculus in solving definite integrals?
3. Why is $\sec^2(x)$ the derivative of $\tan(x)$?
4. How would you handle this problem if no substitution was immediately obvious?
5. What happens if the limits of integration are reversed?

Tip: Recognizing derivatives of common trigonometric functions is key to efficient problem solving in calculus.