Solve the following integration
`int cos x^(@)dx`

Answer

To solve the integral \( \int \cos x \, dx \) where \( x \) is in degrees, we can follow these steps:

### Step 1: Understand the conversion from degrees to radians
Since the standard form for trigonometric functions in calculus is in radians, we need to convert degrees to radians. We know that:
\[ 180^\circ = \pi \text{ radians} \]
Thus, to convert \( x \) degrees to radians, we use the conversion factor:
\[ x \text{ degrees} = \frac{\pi x}{180} \text{ radians} \]

### Step 2: Substitute the conversion into the integral
We denote the integral as:
\[ I = \int \cos x \, dx \]
Substituting the conversion into the integral gives:
\[ I = \int \cos\left(\frac{\pi x}{180}\right) \, dx \]

### Step 3: Use the integral of cosine
The integral of \( \cos(kx) \) is given by:
\[ \int \cos(kx) \, dx = \frac{1}{k} \sin(kx) + C \]
In our case, \( k = \frac{\pi}{180} \). Therefore, we have:
\[ I = \int \cos\left(\frac{\pi x}{180}\right) \, dx = \frac{1}{\frac{\pi}{180}} \sin\left(\frac{\pi x}{180}\right) + C \]

### Step 4: Simplify the expression
Calculating \( \frac{1}{\frac{\pi}{180}} \) gives:
\[ \frac{180}{\pi} \]
Thus, we can write:
\[ I = \frac{180}{\pi} \sin\left(\frac{\pi x}{180}\right) + C \]

### Step 5: Convert back to degrees
Since \( \frac{\pi x}{180} \) can be rewritten as \( \sin(x^\circ) \), we can express the final answer as:
\[ I = \frac{180}{\pi} \sin(x^\circ) + C \]

### Final Answer
\[ \int \cos x \, dx = \frac{180}{\pi} \sin(x^\circ) + C \]

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