`"If " int x e^(x) cosx dx=ae^(x)(b(1-x)sinx+cx cosx)+d,` then

Options

A. `a=1,b=1,c=-1`
B. `a=(1)/(2),b=-1,c=1`
C. `a=1,b=-1,c=1`
D. `a=(1)/(2),b=1,c=-1`

Explanation

To solve the integral \( \int x e^x \cos x \, dx \) and find the values of \( a, b, c \) in the expression

\[ \int x e^x \cos x \, dx = a e^x \left( b(1-x) \sin x + c x \cos x \right) + d, \]

we will differentiate both sides with respect to \( x \).

### Step 1: Differentiate both sides

Differentiating the left-hand side:

\[ \frac{d}{dx} \left( \int x e^x \cos x \, dx \right) = x e^x \cos x. \]

Now, differentiate the right-hand side:

\[ \frac{d}{dx} \left( a e^x \left( b(1-x) \sin x + c x \cos x \right) + d \right). \]

Using the product rule, we get:

\[ \frac{d}{dx} \left( a e^x \right) \left( b(1-x) \sin x + c x \cos x \right) + a e^x \frac{d}{dx} \left( b(1-x) \sin x + c x \cos x \right). \]

### Step 2: Apply the product rule

The derivative of \( a e^x \) is \( a e^x \). Now we need to differentiate \( b(1-x) \sin x + c x \cos x \):

1. For \( b(1-x) \sin x \):
- Using the product rule:
\[ \frac{d}{dx} \left( b(1-x) \sin x \right) = b(-\sin x + (1-x) \cos x). \]

2. For \( c x \cos x \):
- Again using the product rule:
\[ \frac{d}{dx} \left( c x \cos x \right) = c(\cos x - x \sin x). \]

Combining these results, we have:

\[ \frac{d}{dx} \left( b(1-x) \sin x + c x \cos x \right) = b(-\sin x + (1-x) \cos x) + c(\cos x - x \sin x). \]

### Step 3: Combine and simplify

Now substituting back into the derivative of the right-hand side:

\[ x e^x \cos x = a e^x \left( b(1-x) \sin x + c x \cos x \right) + a e^x \left( b(-\sin x + (1-x) \cos x) + c(\cos x - x \sin x) \right). \]

### Step 4: Equate coefficients

Now we will equate coefficients of \( e^x \sin x \) and \( e^x \cos x \) from both sides.

1. Coefficient of \( e^x \sin x \):
\[ 0 = ab - ac \quad \text{(from the left side)} \]

2. Coefficient of \( e^x \cos x \):
\[ 1 = ab + ac \quad \text{(from the left side)} \]

3. Coefficient of \( e^x \):
\[ 0 = -ab - ac \quad \text{(from the left side)} \]

### Step 5: Solve the equations

From the equations:

1. \( ab - ac = 0 \) implies \( a(b - c) = 0 \).
2. \( ab + ac = 1 \).
3. \( -ab - ac = 0 \) implies \( ab + ac = 0 \).

From \( ab + ac = 0 \) and \( ab + ac = 1 \), we can conclude that \( ab + ac = 0 \) is not possible unless both \( a \) and \( b \) are zero, which contradicts our previous equations.

### Step 6: Conclusion

By solving these equations, we can find the values of \( a, b, c \) that satisfy the original integral.