`Evaluate Lim_(n->oo)n^2 int_(-1/n)^(1/n) (2006sinx+2007cosx) |x|dx`

Answer

To evaluate the limit

\[ \lim_{n \to \infty} n^2 \int_{-\frac{1}{n}}^{\frac{1}{n}} (2006 \sin x + 2007 \cos x) |x| \, dx, \]

we will break down the problem step by step.

### Step 1: Break the Integral into Two Parts
We can rewrite the integral as follows:

\[ I = \int_{-\frac{1}{n}}^{\frac{1}{n}} (2006 \sin x + 2007 \cos x) |x| \, dx = \int_{-\frac{1}{n}}^{\frac{1}{n}} 2006 \sin x |x| \, dx + \int_{-\frac{1}{n}}^{\frac{1}{n}} 2007 \cos x |x| \, dx. \]

### Step 2: Analyze the First Integral
The first integral,

\[ \int_{-\frac{1}{n}}^{\frac{1}{n}} 2006 \sin x |x| \, dx, \]

is an odd function because \(\sin(-x) = -\sin(x)\) and \(|x|\) is even. Therefore, the integral over a symmetric interval around zero will be zero:

\[ \int_{-\frac{1}{n}}^{\frac{1}{n}} 2006 \sin x |x| \, dx = 0. \]

### Step 3: Analyze the Second Integral
Now we consider the second integral:

\[ \int_{-\frac{1}{n}}^{\frac{1}{n}} 2007 \cos x |x| \, dx. \]

This function is even because \(\cos(-x) = \cos(x)\) and \(|x|\) is also even. Therefore, we can simplify it:

\[ \int_{-\frac{1}{n}}^{\frac{1}{n}} 2007 \cos x |x| \, dx = 2 \int_{0}^{\frac{1}{n}} 2007 \cos x x \, dx. \]

### Step 4: Substitute Back into the Limit
Now substituting back into \(I\):

\[ I = 2 \int_{0}^{\frac{1}{n}} 2007 x \cos x \, dx. \]

Thus, we have:

\[ \lim_{n \to \infty} n^2 I = \lim_{n \to \infty} n^2 \cdot 2 \int_{0}^{\frac{1}{n}} 2007 x \cos x \, dx. \]

### Step 5: Change of Variable
Let \(u = nx\), then \(x = \frac{u}{n}\) and \(dx = \frac{du}{n}\). The limits change from \(0\) to \(1\):

\[ \int_{0}^{\frac{1}{n}} x \cos x \, dx = \int_{0}^{1} \frac{u}{n} \cos\left(\frac{u}{n}\right) \frac{du}{n} = \frac{1}{n^2} \int_{0}^{1} u \cos\left(\frac{u}{n}\right) \, du. \]

### Step 6: Evaluate the Limit
Now substituting this back into the limit:

\[ \lim_{n \to \infty} n^2 \cdot 2 \cdot 2007 \cdot \frac{1}{n^2} \int_{0}^{1} u \cos\left(\frac{u}{n}\right) \, du = 4014 \lim_{n \to \infty} \int_{0}^{1} u \cos\left(\frac{u}{n}\right) \, du. \]

As \(n \to \infty\), \(\cos\left(\frac{u}{n}\right) \to 1\):

\[ \lim_{n \to \infty} \int_{0}^{1} u \cos\left(\frac{u}{n}\right) \, du = \int_{0}^{1} u \, du = \left[\frac{u^2}{2}\right]_{0}^{1} = \frac{1}{2}. \]

### Step 7: Final Calculation
Thus, we have:

\[ 4014 \cdot \frac{1}{2} = 2007. \]

### Final Answer
The final value is

\[ \boxed{2007}. \]