If `f(x)=int_0^x("cos"(sint)+"cos"(cost)dt`, then `f(x+pi)` is (a) `f(x)+f(pi)` (b) `f(x)+2(pi)` (c) `f(x)+f(pi/2)` (d) `f(x)+2f(pi/2)`

Options

A. `f(x)+f(pi)`
B. `f(x)+2f(pi)`
C. `f(x)+f((pi)/2)`
D. `f(x)+2f((pi)/2)`

Explanation

To solve the problem, we need to evaluate \( f(x + \pi) \) where

\[ f(x) = \int_0^x \left( \cos(\sin t) + \cos(\cos t) \right) dt. \]

### Step 1: Write down the expression for \( f(x + \pi) \)

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

\[ f(x + \pi) = \int_0^{x + \pi} \left( \cos(\sin t) + \cos(\cos t) \right) dt. \]

### Step 2: Break the integral into two parts

We can split the integral into two parts:

\[ f(x + \pi) = \int_0^{x} \left( \cos(\sin t) + \cos(\cos t) \right) dt + \int_x^{x + \pi} \left( \cos(\sin t) + \cos(\cos t) \right) dt. \]

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

\[ f(x + \pi) = f(x) + \int_x^{x + \pi} \left( \cos(\sin t) + \cos(\cos t) \right) dt. \]

### Step 3: Evaluate the second integral

Now we need to evaluate the second integral:

\[ \int_x^{x + \pi} \left( \cos(\sin t) + \cos(\cos t) \right) dt. \]

Using the periodic properties of the integrand, we can analyze this integral. The functions \( \cos(\sin t) \) and \( \cos(\cos t) \) are periodic with period \( 2\pi \). Therefore, we can rewrite the integral:

\[ \int_x^{x + \pi} \left( \cos(\sin t) + \cos(\cos t) \right) dt = \int_0^{\pi} \left( \cos(\sin u) + \cos(\cos u) \right) du, \]

where \( u = t - x \) and the limits change accordingly.

### Step 4: Recognize the integral as a constant

Let’s denote:

\[ C = \int_0^{\pi} \left( \cos(\sin t) + \cos(\cos t) \right) dt. \]

Thus, we have:

\[ f(x + \pi) = f(x) + C. \]

### Step 5: Evaluate \( C \)

Next, we need to evaluate \( C \):

\[ C = \int_0^{\pi} \left( \cos(\sin t) + \cos(\cos t) \right) dt. \]

This integral evaluates to a constant value, which we denote as \( f(\pi) \).

### Final Step: Combine results

Now we can express \( f(x + \pi) \):

\[ f(x + \pi) = f(x) + f(\pi). \]

Thus, the answer is:

\[ f(x + \pi) = f(x) + f(\pi). \]

### Conclusion

The correct answer is:

(a) \( f(x) + f(\pi) \).