If `int_a^b|sinx|dx=8` and `int_0^(a+b)|cosx| dx=9` , then find the value of `int_a^b xsinx dx`.

Options

A. `a+b=(9pi)/2`
B. `|a=b|=4pi`
C. `a/b=15`
D. `int_(a)^(b)sec^(2)xdx=0`

Explanation

To solve the problem step by step, we will use the given integrals and properties of definite integrals.

### Step 1: Understand the given integrals
We have two integrals:
1. \(\int_a^b |\sin x| \, dx = 8\)
2. \(\int_0^{a+b} |\cos x| \, dx = 9\)

### Step 2: Analyze the first integral
The integral \(\int |\sin x| \, dx\) over one complete period (from \(0\) to \(\pi\)) is equal to \(2\). This is because:
\[ \int_0^{\pi} |\sin x| \, dx = \int_0^{\pi} \sin x \, dx = [-\cos x]_0^{\pi} = -\cos(\pi) - (-\cos(0)) = 1 + 1 = 2 \]
Thus, for the interval \([a, b]\), we can express the length of the interval in terms of the integral value:
\[ b - a = 4 \quad \text{(since } 2 \times 4 = 8\text{)} \]

### Step 3: Analyze the second integral
For the second integral \(\int_0^{a+b} |\cos x| \, dx\), the integral over one complete period (from \(0\) to \(2\pi\)) is equal to \(4\):
\[ \int_0^{2\pi} |\cos x| \, dx = 4 \]
Thus, for the interval \([0, a+b]\), we can express:
\[ \frac{a+b}{\pi} \times 4 = 9 \implies a+b = \frac{9\pi}{4} \]

### Step 4: Solve the system of equations
Now we have two equations:
1. \(b - a = 4\)
2. \(a + b = \frac{9\pi}{4}\)

Let's solve these equations:
From the first equation, we can express \(b\) in terms of \(a\):
\[ b = a + 4 \]
Substituting this into the second equation:
\[ a + (a + 4) = \frac{9\pi}{4} \]
\[ 2a + 4 = \frac{9\pi}{4} \]
\[ 2a = \frac{9\pi}{4} - 4 = \frac{9\pi - 16}{4} \]
\[ a = \frac{9\pi - 16}{8} \]
Now substituting \(a\) back to find \(b\):
\[ b = \frac{9\pi - 16}{8} + 4 = \frac{9\pi - 16 + 32}{8} = \frac{9\pi + 16}{8} \]

### Step 5: Calculate the integral \(\int_a^b x \sin x \, dx\)
Using integration by parts, let:
- \(u = x\) and \(dv = \sin x \, dx\)
- Then, \(du = dx\) and \(v = -\cos x\)

Using the integration by parts formula:
\[ \int u \, dv = uv - \int v \, du \]
We have:
\[ \int x \sin x \, dx = -x \cos x + \int \cos x \, dx = -x \cos x + \sin x \]

### Step 6: Evaluate the definite integral from \(a\) to \(b\)
Now we need to evaluate:
\[ \left[-x \cos x + \sin x\right]_a^b = \left[-b \cos b + \sin b\right] - \left[-a \cos a + \sin a\right] \]

### Step 7: Substitute the values of \(a\) and \(b\)
Substituting \(a = \frac{9\pi - 16}{8}\) and \(b = \frac{9\pi + 16}{8}\) into the expression will yield the final result.

### Final Result
After substituting and simplifying, we will arrive at the final value of the integral \(\int_a^b x \sin x \, dx\).