It appears you have uploaded an image containing a mathematical problem. Let me process the content of the image for better clarity and provide a solution.The problem in the image involves solving an integral using substitution, specifically $u = \cos x$. Let's break this down:

The given integral is: $\int \cos x \, dx$

Step 1: Substitution

Let: $u = \cos x$

Step 2: Differentiate $u$

Differentiating $u$ with respect to $x$: $\frac{du}{dx} = -\sin x \quad \text{or} \quad du = -\sin x \, dx$

Step 3: Substitute in the integral

Using the substitution, the integral becomes: $\int \cos x \, dx = \int u \cdot (-du)$

Simplify: $\int \cos x \, dx = -\int u \, du$

Step 4: Solve the new integral

The integral of $u$ with respect to $u$ is: $-\int u \, du = -\frac{u^2}{2} + C$

Step 5: Substitute back $u = \cos x$

Returning to the original variable: $-\frac{u^2}{2} + C = -\frac{\cos^2 x}{2} + C$

Final Answer:

$\int \cos x \, dx = -\frac{\cos^2 x}{2} + C$

Would you like more details, or do you have additional questions? Here are some related questions to explore:

1. How is substitution chosen in integrals?
2. What are other common substitutions for trigonometric integrals?
3. Can you integrate using different techniques besides substitution?
4. How do we verify the correctness of an integral solution?
5. How can this method extend to more complex trigonometric integrals?

Tip: Always double-check the derivative of your solution to confirm its accuracy!