`intcos x^(@)dx=`

Options

A. `sinx^(@)`
B. `sinx`
C. `(180)/(pi)sin((pi)/(180)x)`
D. `(180)/(pi)sinx`

Answer

To solve the integral \( \int \cos x^\circ \, dx \), we will follow these steps:

### Step 1: Convert Degrees to Radians
Since the integral involves \( \cos x^\circ \), we need to convert the angle from degrees to radians. The conversion formula is:
\[ \text{radians} = \frac{\pi}{180} \times \text{degrees} \]
Thus, we can express \( x^\circ \) in radians as:
\[ x^\circ = \frac{\pi}{180} x \]

### Step 2: Rewrite the Integral
Now we can rewrite the integral using the radian measure:
\[ \int \cos x^\circ \, dx = \int \cos\left(\frac{\pi}{180} x\right) \, dx \]

### Step 3: Perform the Integration
To integrate \( \cos\left(\frac{\pi}{180} x\right) \), we use the formula for the integral of cosine:
\[ \int \cos(kx) \, dx = \frac{1}{k} \sin(kx) + C \]
In our case, \( k = \frac{\pi}{180} \). Therefore, we have:
\[ \int \cos\left(\frac{\pi}{180} x\right) \, dx = \frac{1}{\frac{\pi}{180}} \sin\left(\frac{\pi}{180} x\right) + C \]

### Step 4: Simplify the Result
The reciprocal of \( \frac{\pi}{180} \) is \( \frac{180}{\pi} \). Thus, we can write:
\[ \int \cos\left(\frac{\pi}{180} x\right) \, dx = \frac{180}{\pi} \sin\left(\frac{\pi}{180} x\right) + C \]

### Final Result
The final result of the integration is:
\[ \int \cos x^\circ \, dx = \frac{180}{\pi} \sin\left(\frac{\pi}{180} x\right) + C \]