The task in the image is to prove that the function $F(x) = 4\sqrt{x} + \sin(5x)$ is a primitive (antiderivative) for the function $f(x) = \frac{2}{\sqrt{x}} + 5\cos(5x)$, with $x \in (0, +\infty)$.

Solution:

To prove that $F(x)$ is a primitive of $f(x)$, we need to show that the derivative of $F(x)$ is equal to $f(x)$.

Step 1: Differentiate $F(x) = 4\sqrt{x} + \sin(5x)$

• $\frac{d}{dx} \left( 4\sqrt{x} \right) = 4 \cdot \frac{d}{dx} \left( x^{1/2} \right) = 4 \cdot \frac{1}{2} x^{-1/2} = \frac{2}{\sqrt{x}}$
• $\frac{d}{dx} \left( \sin(5x) \right) = 5\cos(5x)$

So, $F'(x) = \frac{2}{\sqrt{x}} + 5\cos(5x)$

Step 2: Compare $F'(x)$ with $f(x)$

We are given that $f(x) = \frac{2}{\sqrt{x}} + 5\cos(5x)$, which is exactly the derivative $F'(x)$.

Thus, $F'(x) = f(x)$, proving that $F(x) = 4\sqrt{x} + \sin(5x)$ is indeed a primitive function for $f(x) = \frac{2}{\sqrt{x}} + 5\cos(5x)$.

Conclusion:

We have shown that the derivative of $F(x)$ equals $f(x)$, meaning $F(x)$ is the antiderivative of $f(x)$.

Would you like more details or have any further questions?

Here are 5 related questions to expand on this topic:

1. How do you find the general solution for a differential equation involving trigonometric functions?
2. What are the properties of antiderivatives and how do they differ from definite integrals?
3. Can you explain how the chain rule is applied in differentiating composite functions like $\sin(5x)$?
4. How does integration work for functions involving square roots like $4\sqrt{x}$?
5. What are some typical functions where trigonometric and algebraic terms appear together?

Tip:

When differentiating composite functions (like $\sin(5x)$), remember to apply the chain rule, multiplying by the derivative of the inner function.