If `int_(0)^(pi/2) f ( sin2 x ) sin x dx = A int_(0)^(pi/4) f ( cos 2 x ) cos x dx` then the value of `A` is `( sqrt2 = 1.41)`

Answer

To solve the problem, we need to find the value of \( A \) in the equation:

\[ \int_{0}^{\frac{\pi}{2}} f(\sin 2x) \sin x \, dx = A \int_{0}^{\frac{\pi}{4}} f(\cos 2x) \cos x \, dx \]

### Step-by-Step Solution:

1. **Define the Left-Hand Side Integral**:
Let
\[ I = \int_{0}^{\frac{\pi}{2}} f(\sin 2x) \sin x \, dx \]

2. **Use the Property of Definite Integrals**:
We can use the property of definite integrals which states:
\[ \int_{a}^{b} f(x) \, dx = \int_{a}^{b} f(b + a - x) \, dx \]
In our case, we will replace \( x \) with \( \frac{\pi}{2} - x \):
\[ I = \int_{0}^{\frac{\pi}{2}} f(\sin(2(\frac{\pi}{2} - x))) \sin(\frac{\pi}{2} - x) \, dx \]

3. **Simplify the Integral**:
We know that:
\[ \sin(2(\frac{\pi}{2} - x)) = \sin(\pi - 2x) = \sin(2x) \]
and
\[ \sin(\frac{\pi}{2} - x) = \cos x \]
So we can rewrite the integral:
\[ I = \int_{0}^{\frac{\pi}{2}} f(\sin 2x) \cos x \, dx \]

4. **Combine the Two Expressions for \( I \)**:
Now we have two expressions for \( I \):
\[ I = \int_{0}^{\frac{\pi}{2}} f(\sin 2x) \sin x \, dx \]
\[ I = \int_{0}^{\frac{\pi}{2}} f(\sin 2x) \cos x \, dx \]
Adding these two equations gives:
\[ 2I = \int_{0}^{\frac{\pi}{2}} f(\sin 2x) (\sin x + \cos x) \, dx \]

5. **Factor Out the Common Terms**:
We can factor out \( \frac{1}{\sqrt{2}} \) from \( \sin x + \cos x \):
\[ \sin x + \cos x = \sqrt{2} \sin\left(x + \frac{\pi}{4}\right) \]
Thus,
\[ 2I = \sqrt{2} \int_{0}^{\frac{\pi}{2}} f(\sin 2x) \sin\left(x + \frac{\pi}{4}\right) \, dx \]

6. **Change of Variables**:
Now, we change variables in the right-hand side integral:
Let \( t = x - \frac{\pi}{4} \), then \( x = t + \frac{\pi}{4} \) and \( dx = dt \). The limits change accordingly:
When \( x = 0 \), \( t = -\frac{\pi}{4} \) and when \( x = \frac{\pi}{2} \), \( t = \frac{\pi}{4} \).

7. **Final Expression**:
After substituting, we can express the integral in terms of \( t \):
\[ 2I = \sqrt{2} \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} f(\sin(2(t + \frac{\pi}{4}))) \cos t \, dt \]
Using the properties of sine and cosine, we can simplify this integral.

8. **Equate the Two Integrals**:
Now we equate the two integrals:
\[ I = A \int_{0}^{\frac{\pi}{4}} f(\cos 2x) \cos x \, dx \]
From our calculations, we find that:
\[ A = \sqrt{2} \]

### Conclusion:
Thus, the value of \( A \) is:
\[ A = \sqrt{2} \approx 1.41 \]