If `I_(10) = int_0^(pi//2)x^(10)` sin x dx then the value of `I_(10) + 90I_8` is

Options

A. `9(pi/2)^(6)`
B. `(pi/2)^(9)`
C. `10(pi/2)^(9)`
D. `9(pi/2)^(9)`

Answer

To solve the integral \( I_{10} = \int_0^{\frac{\pi}{2}} x^{10} \sin x \, dx \) and find the value of \( I_{10} + 90 I_8 \), we will use integration by parts.

### Step 1: Apply Integration by Parts
We use the integration by parts formula:
\[ \int u \, dv = uv - \int v \, du \]
Here, we choose:
- \( u = x^{10} \) (algebraic function)
- \( dv = \sin x \, dx \)

Now, we need to find \( du \) and \( v \):
- \( du = 10x^9 \, dx \)
- \( v = -\cos x \)

### Step 2: Substitute into the Integration by Parts Formula
Applying the integration by parts formula:
\[ I_{10} = \left[ -x^{10} \cos x \right]_0^{\frac{\pi}{2}} - \int_0^{\frac{\pi}{2}} -\cos x \cdot 10x^9 \, dx \]

### Step 3: Evaluate the Boundary Terms
Now, we evaluate the boundary term:
\[ \left[ -x^{10} \cos x \right]_0^{\frac{\pi}{2}} = -\left( \left( \frac{\pi}{2} \right)^{10} \cos\left( \frac{\pi}{2} \right) - 0^{10} \cos(0) \right) \]
Since \( \cos\left( \frac{\pi}{2} \right) = 0 \) and \( \cos(0) = 1 \):
\[ = -\left( 0 - 0 \right) = 0 \]

### Step 4: Simplify the Integral
Thus, we have:
\[ I_{10} = 0 + 10 \int_0^{\frac{\pi}{2}} x^9 \cos x \, dx \]
Let \( I_9 = \int_0^{\frac{\pi}{2}} x^9 \cos x \, dx \), so:
\[ I_{10} = 10 I_9 \]

### Step 5: Apply Integration by Parts Again for \( I_9 \)
Now we apply integration by parts to \( I_9 \):
- Let \( u = x^9 \) and \( dv = \cos x \, dx \)
- Then \( du = 9x^8 \, dx \) and \( v = \sin x \)

Using the integration by parts formula again:
\[ I_9 = \left[ x^9 \sin x \right]_0^{\frac{\pi}{2}} - \int_0^{\frac{\pi}{2}} \sin x \cdot 9x^8 \, dx \]
Evaluating the boundary term:
\[ \left[ x^9 \sin x \right]_0^{\frac{\pi}{2}} = \left( \frac{\pi}{2} \right)^9 \cdot 1 - 0 = \left( \frac{\pi}{2} \right)^9 \]
Thus:
\[ I_9 = \left( \frac{\pi}{2} \right)^9 - 9 I_8 \]

### Step 6: Substitute Back into \( I_{10} \)
Now substituting \( I_9 \) back into the equation for \( I_{10} \):
\[ I_{10} = 10 \left( \left( \frac{\pi}{2} \right)^9 - 9 I_8 \right) \]
This simplifies to:
\[ I_{10} = 10 \left( \frac{\pi}{2} \right)^9 - 90 I_8 \]

### Step 7: Rearranging the Equation
Now we want to find \( I_{10} + 90 I_8 \):
\[ I_{10} + 90 I_8 = 10 \left( \frac{\pi}{2} \right)^9 \]

### Final Answer
Thus, the value of \( I_{10} + 90 I_8 \) is:
\[ \boxed{10 \left( \frac{\pi}{2} \right)^9} \]