The valueof `intf(x)g^('')(x)-f^('')(x)g(x)+C`

Options

A. `(f(x)/g^(')x)+C`
B. `f^(')(x)g(x)-f(x)g^(')(x)+C`
C. `f(x)g^(')(x)-f^(')(x)g(x)+C`
D. `f(x)g^(')(x)+f^(')(x)g^(')(x)+C`

Answer

To solve the integral \( I = \int \left( f(x) g''(x) - f''(x) g(x) \right) \, dx + C \), we will break it down step by step.

### Step 1: Separate the Integral
We can separate the integral into two parts:
\[ I = \int f(x) g''(x) \, dx - \int f''(x) g(x) \, dx + C \]

### Step 2: Apply Integration by Parts to the First Integral
For the first integral \( \int f(x) g''(x) \, dx \), we will use integration by parts. Let:
- \( u = f(x) \) (thus \( du = f'(x) \, dx \))
- \( dv = g''(x) \, dx \) (thus \( v = g'(x) \))

Using the integration by parts formula:
\[ \int u \, dv = uv - \int v \, du \]
we get:
\[ \int f(x) g''(x) \, dx = f(x) g'(x) - \int g'(x) f'(x) \, dx \]

### Step 3: Substitute Back into the Integral
Now substituting back into the integral \( I \):
\[ I = \left( f(x) g'(x) - \int g'(x) f'(x) \, dx \right) - \int f''(x) g(x) \, dx + C \]

### Step 4: Apply Integration by Parts to the Second Integral
Now we focus on the second integral \( \int f''(x) g(x) \, dx \). Again, we will use integration by parts. Let:
- \( u = g(x) \) (thus \( du = g'(x) \, dx \))
- \( dv = f''(x) \, dx \) (thus \( v = f'(x) \))

Using the integration by parts formula again:
\[ \int g(x) f''(x) \, dx = g(x) f'(x) - \int f'(x) g'(x) \, dx \]

### Step 5: Substitute Back into the Integral
Substituting this back into our expression for \( I \):
\[ I = f(x) g'(x) - \int g'(x) f'(x) \, dx - \left( g(x) f'(x) - \int f'(x) g'(x) \, dx \right) + C \]

### Step 6: Simplify the Expression
Now, simplifying the expression:
\[ I = f(x) g'(x) - g(x) f'(x) + \int f'(x) g'(x) \, dx - \int g'(x) f'(x) \, dx + C \]
Notice that the two integral terms \( \int f'(x) g'(x) \, dx \) and \( -\int g'(x) f'(x) \, dx \) cancel each other out.

### Final Result
Thus, we are left with:
\[ I = f(x) g'(x) - g(x) f'(x) + C \]

### Conclusion
The final value of the integral is:
\[ \int \left( f(x) g''(x) - f''(x) g(x) \right) \, dx + C = f(x) g'(x) - g(x) f'(x) + C \]