`525int_0^(pi/2)sin2x(cosx)^(11/2)(1+(cosx)^(5/2))^(1/2)dx`

Options

A. `64+176sqrt2`
B. `176sqrt2-64`
C. `64-128sqrt2`
D. `64+128sqrt2`

Answer

To solve the integral

\[ 525 \int_0^{\frac{\pi}{2}} \sin(2x) (\cos x)^{\frac{11}{2}} \left(1 + (\cos x)^{\frac{5}{2}}\right)^{\frac{1}{2}} dx, \]

we can follow these steps:

### Step 1: Simplify the Integral
Using the identity for \(\sin(2x) = 2 \sin x \cos x\), we can rewrite the integral:

\[ = 525 \int_0^{\frac{\pi}{2}} 2 \sin x \cos x (\cos x)^{\frac{11}{2}} \left(1 + (\cos x)^{\frac{5}{2}}\right)^{\frac{1}{2}} dx. \]

This simplifies to:

\[ = 1050 \int_0^{\frac{\pi}{2}} \sin x (\cos x)^{\frac{13}{2}} \left(1 + (\cos x)^{\frac{5}{2}}\right)^{\frac{1}{2}} dx. \]

### Step 2: Use Substitution
Let \( t = 1 + (\cos x)^{\frac{5}{2}} \). Then, we need to find \( dt \):

\[ dt = \frac{5}{2} (\cos x)^{\frac{3}{2}} (-\sin x) dx. \]

From this, we can express \( \sin x dx \) in terms of \( dt \):

\[ \sin x dx = -\frac{2}{5} (\cos x)^{-\frac{3}{2}} dt. \]

### Step 3: Change the Limits of Integration
When \( x = 0 \), \( t = 1 + 1^{\frac{5}{2}} = 2 \).

When \( x = \frac{\pi}{2} \), \( t = 1 + 0 = 1 \).

Thus, the limits change from \( 0 \) to \( \frac{\pi}{2} \) to \( 2 \) to \( 1 \).

### Step 4: Rewrite the Integral
Substituting into the integral, we have:

\[ = 1050 \int_2^1 \left(-\frac{2}{5} (\cos x)^{-\frac{3}{2}}\right) (\cos x)^{\frac{13}{2}} \sqrt{t} dt. \]

This simplifies to:

\[ = 1050 \cdot \frac{2}{5} \int_1^2 \sqrt{t} t^{\frac{5}{2}} dt. \]

### Step 5: Evaluate the Integral
Now, we need to evaluate:

\[ = 420 \int_1^2 t^{3} dt. \]

Calculating the integral:

\[ = 420 \left[ \frac{t^{4}}{4} \right]_1^2 = 420 \left( \frac{16}{4} - \frac{1}{4} \right) = 420 \left( 4 - \frac{1}{4} \right) = 420 \left( \frac{16 - 1}{4} \right) = 420 \cdot \frac{15}{4} = 1575. \]

### Final Answer
Thus, the value of the integral is:

\[ \boxed{1575}. \]