Consider the function `f(x)=int_(0)^(x)(5ln(1+t^(2))-10t tan^(-1)t+16sint)dt`. Which is not true for `int_(0)^(x)f(t)dt` gt?

Options

A. positive for all `x in (0,1)`
B. increasing for all `x in (0,1)`
C. non-monotonic for all `x in (0,1)`
D. none of these

Explanation

To solve the problem, we need to analyze the function given by the integral and its properties. Let's break it down step by step.

### Step 1: Define the function
We are given the function:
\[ f(x) = \int_0^x \left( 5 \ln(1 + t^2) - 10t \tan^{-1}(t) + 16 \sin(t) \right) dt \]

### Step 2: Differentiate \( f(x) \)
To analyze the behavior of \( f(x) \), we will find its derivative using the Fundamental Theorem of Calculus:
\[ f'(x) = 5 \ln(1 + x^2) - 10x \tan^{-1}(x) + 16 \sin(x) \]

### Step 3: Find the second derivative \( f''(x) \)
Next, we differentiate \( f'(x) \) to find the second derivative:
\[ f''(x) = \frac{d}{dx} \left( 5 \ln(1 + x^2) \right) - \frac{d}{dx} \left( 10x \tan^{-1}(x) \right) + \frac{d}{dx} \left( 16 \sin(x) \right) \]

Calculating each term:
1. \( \frac{d}{dx} \left( 5 \ln(1 + x^2) \right) = \frac{10x}{1 + x^2} \)
2. Using the product rule for \( 10x \tan^{-1}(x) \):
\[ \frac{d}{dx} \left( 10x \tan^{-1}(x) \right) = 10 \tan^{-1}(x) + \frac{10x}{1 + x^2} \]
3. \( \frac{d}{dx} \left( 16 \sin(x) \right) = 16 \cos(x) \)

Putting it all together:
\[ f''(x) = \frac{10x}{1 + x^2} - \left( 10 \tan^{-1}(x) + \frac{10x}{1 + x^2} \right) + 16 \cos(x) \]
This simplifies to:
\[ f''(x) = -10 \tan^{-1}(x) + 16 \cos(x) \]

### Step 4: Analyze the behavior of \( f''(x) \)
To determine where \( f''(x) \) is positive or negative, we need to analyze the expression:
\[ f''(x) = -10 \tan^{-1}(x) + 16 \cos(x) \]
We need to find intervals where this expression is greater than or less than zero.

### Step 5: Evaluate \( \int_0^x f(t) dt \)
To find the integral \( \int_0^x f(t) dt \), we can apply the Fundamental Theorem of Calculus again:
\[ \int_0^x f(t) dt = F(x) \text{ where } F'(x) = f(x) \]

### Conclusion
The properties of \( f(x) \) and \( f'(x) \) will help us determine which statements about \( \int_0^x f(t) dt \) are true or false.