If `u_(n) = int_(0)^(pi/2) x^(n)sinxdx`, then the value of `u_(10) + 90 u_(8)` is : (a) `9(pi/2)^(8)` (b) `(pi/2)^(9)` (c) `10 (pi/2)^(9)` (d) `9 (pi/2)^(9)`

Options

A. `9(pi/2)^(8)`
B. `(pi/2)^(9)`
C. `10 (pi/2)^(9)`
D. `9 (pi/2)^(9)`

Answer

To solve the problem, we need to evaluate \( u_{10} + 90 u_{8} \) where \( u_n = \int_{0}^{\frac{\pi}{2}} x^n \sin x \, dx \).

### Step-by-Step Solution:

1. **Define \( u_n \)**:
\[ u_n = \int_{0}^{\frac{\pi}{2}} x^n \sin x \, dx \]

2. **Calculate \( u_{10} \)**:
We will use integration by parts to evaluate \( u_{10} \).
Let:
- \( v = \sin x \) (thus \( dv = \cos x \, dx \))
- \( w = x^{10} \) (thus \( dw = 10 x^{9} \, dx \))

Using integration by parts:
\[ u_{10} = \left[ -x^{10} \cos x \right]_{0}^{\frac{\pi}{2}} + \int_{0}^{\frac{\pi}{2}} 10 x^{9} \cos x \, dx \]

Evaluating the boundary term:
\[ -\left( \frac{\pi}{2} \right)^{10} \cos\left( \frac{\pi}{2} \right) + 0 = 0 \]
Thus,
\[ u_{10} = 10 \int_{0}^{\frac{\pi}{2}} x^{9} \cos x \, dx \]

3. **Calculate \( u_{9} \)**:
Now we need to evaluate \( \int_{0}^{\frac{\pi}{2}} x^{9} \cos x \, dx \) using integration by parts again:
Let:
- \( v = \cos x \) (thus \( dv = -\sin x \, dx \))
- \( w = x^{9} \) (thus \( dw = 9 x^{8} \, dx \))

Using integration by parts:
\[ \int_{0}^{\frac{\pi}{2}} x^{9} \cos x \, dx = \left[ x^{9} \sin x \right]_{0}^{\frac{\pi}{2}} - \int_{0}^{\frac{\pi}{2}} 9 x^{8} \sin x \, dx \]

Evaluating the boundary term:
\[ \left( \frac{\pi}{2} \right)^{9} \sin\left( \frac{\pi}{2} \right) - 0 = \left( \frac{\pi}{2} \right)^{9} \]
Thus,
\[ \int_{0}^{\frac{\pi}{2}} x^{9} \cos x \, dx = \left( \frac{\pi}{2} \right)^{9} - 9 \int_{0}^{\frac{\pi}{2}} x^{8} \sin x \, dx = \left( \frac{\pi}{2} \right)^{9} - 9 u_{8} \]

4. **Substituting back into \( u_{10} \)**:
Substituting back into the equation for \( u_{10} \):
\[ u_{10} = 10 \left( \left( \frac{\pi}{2} \right)^{9} - 9 u_{8} \right) = 10 \left( \frac{\pi}{2} \right)^{9} - 90 u_{8} \]

5. **Finding \( u_{10} + 90 u_{8} \)**:
Now we can find \( u_{10} + 90 u_{8} \):
\[ u_{10} + 90 u_{8} = \left( 10 \left( \frac{\pi}{2} \right)^{9} - 90 u_{8} \right) + 90 u_{8} = 10 \left( \frac{\pi}{2} \right)^{9} \]

### Final Answer:
Thus, the value of \( u_{10} + 90 u_{8} \) is:
\[ 10 \left( \frac{\pi}{2} \right)^{9} \]
The correct option is (c) \( 10 \left( \frac{\pi}{2} \right)^{9} \).