If `int_(0)^(t) (bx cos 4x- a sin 4x)/(x^(2)) dx= (a sin 4t)/(t)-1`, where `0 lt t lt (pi)/(4)`, then the value of a, b are equal to

Options

A. `(1)/(4), 1`
B. `-1, 4`
C. 2, 2
D. 2, 4

Answer

To solve the given integral equation

\[ \int_{0}^{t} \frac{b x \cos 4x - a \sin 4x}{x^{2}} \, dx = \frac{a \sin 4t}{t} - 1 \]

for \(0 < t < \frac{\pi}{4}\), we will differentiate both sides with respect to \(t\).

### Step 1: Differentiate both sides

Using the Fundamental Theorem of Calculus on the left-hand side:

\[ \frac{d}{dt} \left( \int_{0}^{t} \frac{b x \cos 4x - a \sin 4x}{x^{2}} \, dx \right) = \frac{b t \cos 4t - a \sin 4t}{t^{2}} \]

Now, differentiate the right-hand side:

\[ \frac{d}{dt} \left( \frac{a \sin 4t}{t} - 1 \right) \]

Using the quotient rule \( \frac{d}{dt} \left( \frac{u}{v} \right) = \frac{v \frac{du}{dt} - u \frac{dv}{dt}}{v^2} \):

Let \( u = a \sin 4t \) and \( v = t \):

\[ \frac{du}{dt} = 4a \cos 4t, \quad \frac{dv}{dt} = 1 \]

Thus,

\[ \frac{d}{dt} \left( \frac{a \sin 4t}{t} \right) = \frac{t(4a \cos 4t) - a \sin 4t}{t^2} = \frac{4a t \cos 4t - a \sin 4t}{t^2} \]

### Step 2: Set the derivatives equal

Now we set the derivatives equal to each other:

\[ \frac{b t \cos 4t - a \sin 4t}{t^{2}} = \frac{4a t \cos 4t - a \sin 4t}{t^{2}} \]

### Step 3: Eliminate the denominators

Since \(t^2\) is positive for \(t > 0\), we can multiply both sides by \(t^2\):

\[ b t \cos 4t - a \sin 4t = 4a t \cos 4t - a \sin 4t \]

### Step 4: Simplify the equation

Now, we can simplify:

\[ b t \cos 4t = 4a t \cos 4t \]

### Step 5: Factor out common terms

Assuming \(t \cos 4t \neq 0\) (which is valid since \(0 < t < \frac{\pi}{4}\)), we can divide both sides by \(t \cos 4t\):

\[ b = 4a \]

### Step 6: Substitute back to find \(a\) and \(b\)

Now, we need to find the values of \(a\) and \(b\). We can substitute \(b = 4a\) back into the original equation.

To find \(a\) and \(b\), we need to analyze the limits or specific values. If we assume \(a = 1\), then:

\[ b = 4 \cdot 1 = 4 \]

### Conclusion

Thus, the values of \(a\) and \(b\) are:

\[ \boxed{(1, 4)} \]