What is `int sin^(2) x dx + int cos^(2) x dx` equal to ?
Where c is an arbitary constant

Options

A. `x + c`
B. `(x^(2))/(2) + c`
C. `x^(2) + c`
D. None of these

Explanation

To solve the integral \( \int \sin^2 x \, dx + \int \cos^2 x \, dx \), we can follow these steps:

### Step 1: Write the integrals separately
We start with the expression:
\[ \int \sin^2 x \, dx + \int \cos^2 x \, dx \]

### Step 2: Use the Pythagorean identity
Recall the Pythagorean identity:
\[ \sin^2 x + \cos^2 x = 1 \]
Using this identity, we can combine the two integrals:
\[ \int \sin^2 x \, dx + \int \cos^2 x \, dx = \int (\sin^2 x + \cos^2 x) \, dx = \int 1 \, dx \]

### Step 3: Integrate
Now we can integrate:
\[ \int 1 \, dx = x + C \]
where \( C \) is an arbitrary constant.

### Final Answer
Thus, the value of the original expression is:
\[ \int \sin^2 x \, dx + \int \cos^2 x \, dx = x + C \]

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