`"If" int(dx)/((x^(2)-2x+10)^(2))=A("tan"^(-1)((x-1)/(3))+(f(x))/(x^(2)-2x+10))+C`,where, C is a constant of integration, then

Options

A. `A=(1)/(54),f(x)=3(x-1)`
B. `A=(1)/(54),f(x)=9(x-1)^(2)`
C. `A=(1)/(27),f(x)=9(x-1)^(2)`
D. `A=(1)/(81),f(x)=3(x-1)`

Answer

To solve the given integral problem, we need to evaluate the integral on the left-hand side and compare it with the expression on the right-hand side. Let's break down the solution step by step.

### Step 1: Rewrite the Integral
We start with the integral:
\[ \int \frac{dx}{(x^2 - 2x + 10)^2} \]
We can rewrite \(10\) as \(1 + 9\), so:
\[ x^2 - 2x + 10 = (x^2 - 2x + 1) + 9 = (x - 1)^2 + 3^2 \]
Thus, we have:
\[ \int \frac{dx}{((x - 1)^2 + 3^2)^2} \]

### Step 2: Use Trigonometric Substitution
Let \(x - 1 = 3 \tan \theta\). Then, \(dx = 3 \sec^2 \theta \, d\theta\). Substituting these into the integral gives:
\[ \int \frac{3 \sec^2 \theta \, d\theta}{(3 \tan \theta)^2 + 3^2)^2} = \int \frac{3 \sec^2 \theta \, d\theta}{(9 \tan^2 \theta + 9)^2} = \int \frac{3 \sec^2 \theta \, d\theta}{9^2 (\tan^2 \theta + 1)^2} = \int \frac{3 \sec^2 \theta \, d\theta}{81 \sec^4 \theta} \]
This simplifies to:
\[ \frac{1}{27} \int \cos^2 \theta \, d\theta \]

### Step 3: Integrate
Using the identity \(\cos^2 \theta = \frac{1 + \cos 2\theta}{2}\), we can rewrite the integral:
\[ \frac{1}{27} \int \cos^2 \theta \, d\theta = \frac{1}{27} \cdot \frac{1}{2} \int (1 + \cos 2\theta) \, d\theta = \frac{1}{54} \left( \theta + \frac{1}{2} \sin 2\theta \right) + C \]

### Step 4: Substitute Back
Recall that \(\theta = \tan^{-1}\left(\frac{x - 1}{3}\right)\). Thus, we have:
\[ \int \frac{dx}{(x^2 - 2x + 10)^2} = \frac{1}{54} \tan^{-1}\left(\frac{x - 1}{3}\right) + \text{(some function of x)} + C \]

### Step 5: Compare with the Right-Hand Side
The right-hand side of the original equation is:
\[ A \tan^{-1}\left(\frac{x - 1}{3}\right) + \frac{f(x)}{x^2 - 2x + 10} + C \]
From our integration, we see that:
\[ A = \frac{1}{54} \]
Now we need to find \(f(x)\).

### Step 6: Find \(f(x)\)
From the integration, we can express the remaining part:
\[ \frac{1}{54} \tan^{-1}\left(\frac{x - 1}{3}\right) + \frac{f(x)}{(x - 1)^2 + 3^2} + C \]
To find \(f(x)\), we can differentiate the remaining part of the integral and compare it with the original function. After some algebraic manipulation, we find:
\[ f(x) = 3(x - 1) \]

### Final Answer
Thus, we have:
\[ A = \frac{1}{54}, \quad f(x) = 3(x - 1) \]