Let `f(x)` is differentiable function satisfying `2int_(1)^(2)f(tx) dt = x+2, AA x in R` Then `int_(1)^(2)(8f(8x)-f(x)-21x) dx` equal to

Options

A. 3
B. 5
C. 7
D. 9

Answer

To solve the given problem step by step, we start with the information provided:

Given that:
\[ 2 \int_{1}^{2} f(tx) \, dt = x + 2 \quad \forall x \in \mathbb{R} \]

We need to find:
\[ \int_{1}^{2} (8f(8x) - f(x) - 21x) \, dx \]

### Step 1: Analyze the given equation
From the equation \( 2 \int_{1}^{2} f(tx) \, dt = x + 2 \), we can isolate the integral:
\[ \int_{1}^{2} f(tx) \, dt = \frac{x + 2}{2} \]

### Step 2: Substitute \( x = 8 \)
Now, substituting \( x = 8 \) into the equation:
\[ 2 \int_{1}^{2} f(8t) \, dt = 8 + 2 = 10 \]
Thus,
\[ \int_{1}^{2} f(8t) \, dt = 5 \]

### Step 3: Substitute \( x = 1 \)
Next, substituting \( x = 1 \):
\[ 2 \int_{1}^{2} f(t) \, dt = 1 + 2 = 3 \]
Thus,
\[ \int_{1}^{2} f(t) \, dt = \frac{3}{2} \]

### Step 4: Rewrite the integral we need to evaluate
Now we need to evaluate:
\[ \int_{1}^{2} (8f(8x) - f(x) - 21x) \, dx \]
This can be separated into three integrals:
\[ \int_{1}^{2} 8f(8x) \, dx - \int_{1}^{2} f(x) \, dx - \int_{1}^{2} 21x \, dx \]

### Step 5: Evaluate \( \int_{1}^{2} 8f(8x) \, dx \)
To evaluate \( \int_{1}^{2} 8f(8x) \, dx \), we can use the substitution \( u = 8x \), which gives \( du = 8dx \) or \( dx = \frac{du}{8} \). The limits change from \( x = 1 \) to \( u = 8 \) and from \( x = 2 \) to \( u = 16 \):
\[ \int_{1}^{2} 8f(8x) \, dx = 8 \int_{1}^{2} f(8x) \, dx = \int_{8}^{16} f(u) \frac{du}{8} = \int_{8}^{16} f(u) \, du \]
However, we need to evaluate \( \int_{1}^{2} f(8x) \, dx \) directly:
Using the earlier result, we have:
\[ \int_{1}^{2} f(8x) \, dx = 5 \]
Thus,
\[ 8 \int_{1}^{2} f(8x) \, dx = 8 \cdot 5 = 40 \]

### Step 6: Evaluate \( \int_{1}^{2} f(x) \, dx \)
From earlier, we have:
\[ \int_{1}^{2} f(x) \, dx = \frac{3}{2} \]

### Step 7: Evaluate \( \int_{1}^{2} 21x \, dx \)
Calculating this integral:
\[ \int_{1}^{2} 21x \, dx = 21 \left[ \frac{x^2}{2} \right]_{1}^{2} = 21 \left( \frac{4}{2} - \frac{1}{2} \right) = 21 \cdot \frac{3}{2} = \frac{63}{2} \]

### Step 8: Combine all parts
Now we can combine all parts:
\[ \int_{1}^{2} (8f(8x) - f(x) - 21x) \, dx = 40 - \frac{3}{2} - \frac{63}{2} \]
Calculating:
\[ = 40 - \frac{66}{2} = 40 - 33 = 7 \]

### Final Answer
Thus, the value of the integral is:
\[ \boxed{7} \]