If `g(x)=int_(0)^(x)cos^(4)t dt,` then `g(x+pi)` equals

Options

A. `(g(x))/(g(pi))`
B. `g(x)+g(pi)`
C. `g(x)-g(pi)`
D. `g(x).g(pi)`

Explanation

To solve the problem, we need to find \( g(x + \pi) \) given that

\[ g(x) = \int_0^x \cos^4 t \, dt. \]

### Step 1: Express \( g(x + \pi) \)

We start by substituting \( x + \pi \) into the function \( g \):

\[ g(x + \pi) = \int_0^{x + \pi} \cos^4 t \, dt. \]

### Step 2: Split the Integral

We can split the integral from \( 0 \) to \( x + \pi \) into two parts:

\[ g(x + \pi) = \int_0^{\pi} \cos^4 t \, dt + \int_{\pi}^{x + \pi} \cos^4 t \, dt. \]

### Step 3: Change of Variable in the Second Integral

For the second integral, we can make a change of variable. Let \( u = t - \pi \). Then \( dt = du \) and when \( t = \pi \), \( u = 0 \), and when \( t = x + \pi \), \( u = x \). Thus, we have:

\[ \int_{\pi}^{x + \pi} \cos^4 t \, dt = \int_0^x \cos^4(u + \pi) \, du. \]

### Step 4: Simplify \( \cos^4(u + \pi) \)

Using the property of cosine, we know that:

\[ \cos(u + \pi) = -\cos u. \]

Therefore,

\[ \cos^4(u + \pi) = (-\cos u)^4 = \cos^4 u. \]

### Step 5: Substitute Back

Now substituting back into the integral, we get:

\[ \int_{\pi}^{x + \pi} \cos^4 t \, dt = \int_0^x \cos^4 u \, du = g(x). \]

### Step 6: Combine the Results

Putting it all together, we have:

\[ g(x + \pi) = \int_0^{\pi} \cos^4 t \, dt + g(x). \]

### Step 7: Final Expression

Thus, the final expression for \( g(x + \pi) \) is:

\[ g(x + \pi) = g(x) + \int_0^{\pi} \cos^4 t \, dt. \]