`If g(x)=int_0^x(|sint|+|cost|)dt ,t h e ng(x+(pin)/2)` is equal to, where `n in N ,`
a)`g(x)+g(pi)`
(b) `g(x)+ng((pi)/(n2))`
c)`g(x)+g(pi/2)`
(d) none of these

Options

A. `g(x)+g(pi)`
B. `g(x)+ng((pi)/2)`
C. `g(x)+g((pi)/2)`
D. none of these

Explanation

To solve the problem, we need to evaluate \( g(x + n\frac{\pi}{2}) \) where \( g(x) = \int_0^x (|\sin t| + |\cos t|) dt \).

### Step 1: Write the expression for \( g(x + n\frac{\pi}{2}) \)

We start by substituting \( x + n\frac{\pi}{2} \) into the function \( g \):

\[ g\left(x + n\frac{\pi}{2}\right) = \int_0^{x + n\frac{\pi}{2}} (|\sin t| + |\cos t|) dt \]

### Step 2: Split the integral

We can split the integral into two parts:

\[ g\left(x + n\frac{\pi}{2}\right) = \int_0^x (|\sin t| + |\cos t|) dt + \int_x^{x + n\frac{\pi}{2}} (|\sin t| + |\cos t|) dt \]

The first part is simply \( g(x) \):

\[ g\left(x + n\frac{\pi}{2}\right) = g(x) + \int_x^{x + n\frac{\pi}{2}} (|\sin t| + |\cos t|) dt \]

### Step 3: Evaluate the second integral

Now we need to evaluate the second integral \( \int_x^{x + n\frac{\pi}{2}} (|\sin t| + |\cos t|) dt \). The function \( |\sin t| + |\cos t| \) is periodic with a period of \( \pi \).

### Step 4: Determine the periodicity

Since \( |\sin t| + |\cos t| \) has a period of \( \pi \), we can express the integral over \( n\frac{\pi}{2} \) in terms of its periodic behavior.

For \( n \) even, \( n\frac{\pi}{2} \) covers \( n/2 \) full periods of \( \pi \). For \( n \) odd, it covers \( (n-1)/2 \) full periods and an additional \( \frac{\pi}{2} \).

### Step 5: Calculate the integral

The integral over one full period \( [0, \pi] \) can be computed as follows:

\[ \int_0^{\pi} (|\sin t| + |\cos t|) dt = \int_0^{\pi} |\sin t| dt + \int_0^{\pi} |\cos t| dt \]

Calculating these integrals gives:

\[ \int_0^{\pi} |\sin t| dt = 2 \quad \text{and} \quad \int_0^{\pi} |\cos t| dt = 2 \]

Thus,

\[ \int_0^{\pi} (|\sin t| + |\cos t|) dt = 2 + 2 = 4 \]

### Step 6: Use the periodicity

Using the periodicity, we can write:

\[ \int_x^{x + n\frac{\pi}{2}} (|\sin t| + |\cos t|) dt = n \cdot \int_0^{\pi} (|\sin t| + |\cos t|) dt = n \cdot 4 = 4n \]

### Step 7: Combine the results

Now, we can combine the results:

\[ g\left(x + n\frac{\pi}{2}\right) = g(x) + 4n \]

### Step 8: Identify the correct option

We need to express \( g\left(x + n\frac{\pi}{2}\right) \) in terms of the given options. The closest match is:

\[ g(x) + n g\left(\frac{\pi}{2}\right) \]

Since \( g\left(\frac{\pi}{2}\right) = 4 \), we have:

\[ g\left(x + n\frac{\pi}{2}\right) = g(x) + 4n = g(x) + n g\left(\frac{\pi}{2}\right) \]

Thus, the correct answer is:

**(b) \( g(x) + n g\left(\frac{\pi}{2}\right) \)**