If `int (sin2x-cos 2x)dx=(1)/(sqrt2) sin(2x-a)+b`, then the values of a & b are

Options

A. `a=-(5pi)/(4), b in R`
B. `a=(5pi)/(4), b in R`
C. `a inR, b=(5pi)/(4)`
D. `b=-(5pi)/(4), a in R`

Answer

To solve the integral \( \int (\sin 2x - \cos 2x) \, dx \) and find the values of \( a \) and \( b \) in the equation

\[ \int (\sin 2x - \cos 2x) \, dx = \frac{1}{\sqrt{2}} \sin(2x - a) + b, \]

we will follow these steps:

### Step 1: Integrate the expression
We start by integrating the expression \( \sin 2x - \cos 2x \).

\[ \int (\sin 2x - \cos 2x) \, dx = \int \sin 2x \, dx - \int \cos 2x \, dx. \]

### Step 2: Calculate the integral of \( \sin 2x \)
Using the formula \( \int \sin kx \, dx = -\frac{1}{k} \cos kx + C \):

\[ \int \sin 2x \, dx = -\frac{1}{2} \cos 2x. \]

### Step 3: Calculate the integral of \( \cos 2x \)
Using the formula \( \int \cos kx \, dx = \frac{1}{k} \sin kx + C \):

\[ \int \cos 2x \, dx = \frac{1}{2} \sin 2x. \]

### Step 4: Combine the results
Now, substituting back into our integral:

\[ \int (\sin 2x - \cos 2x) \, dx = -\frac{1}{2} \cos 2x - \frac{1}{2} \sin 2x + C. \]

### Step 5: Factor out \( -\frac{1}{2} \)
We can factor out \( -\frac{1}{2} \):

\[ \int (\sin 2x - \cos 2x) \, dx = -\frac{1}{2} (\cos 2x + \sin 2x) + C. \]

### Step 6: Rewrite using sine addition formula
We can rewrite \( \cos 2x + \sin 2x \) using the sine addition formula. We know:

\[ \cos \theta = \frac{1}{\sqrt{2}}, \quad \sin \theta = \frac{1}{\sqrt{2}} \quad \text{for } \theta = \frac{\pi}{4}. \]

Thus,

\[ \cos 2x + \sin 2x = \sqrt{2} \sin\left(2x + \frac{\pi}{4}\right). \]

So, we have:

\[ -\frac{1}{2} (\cos 2x + \sin 2x) = -\frac{1}{2} \cdot \sqrt{2} \sin\left(2x + \frac{\pi}{4}\right) + C. \]

### Step 7: Simplify the expression
This gives us:

\[ \int (\sin 2x - \cos 2x) \, dx = -\frac{1}{\sqrt{2}} \sin\left(2x + \frac{\pi}{4}\right) + C. \]

### Step 8: Compare with the given equation
Now, we compare this with the given equation:

\[ \frac{1}{\sqrt{2}} \sin(2x - a) + b. \]

From the comparison, we see that:

1. The coefficient of \( \sin \) matches.
2. The angle \( 2x + \frac{\pi}{4} \) corresponds to \( 2x - a \).

This implies:

\[ -a = \frac{\pi}{4} \implies a = -\frac{\pi}{4}. \]

### Step 9: Identify \( b \)
The constant \( b \) corresponds to the integration constant \( C \), which can be any real number.

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

\[ a = -\frac{\pi}{4}, \quad b \in \mathbb{R}. \]