Evaluating integrals dependent on a parameter:
Differentiate I with respect to the parameter within the sign an integrals taking variable of the integrand as constant. Now evaluate the integral so obtained as a function of the parameter then integrate then result of get I. Constant of integration can be computed by giving some arbitrary values to the parameter and the corresponding value of I.
If `int_(0)^(pi)(dx)/((a-cosx))=(pi)/(sqrt(a^(2)-1))`, then the value of `int_(0)^(pi)(dx)/((sqrt(10)-cosx)^3)` is

Options

A. (a) `(pi)/81`
B. (b) `(7pi)/162`
C. (c) `(7pi)/81`
D. (d) none of these

Explanation

To solve the integral \( I = \int_{0}^{\pi} \frac{dx}{(\sqrt{10} - \cos x)^3} \), we will use the given formula and differentiate with respect to the parameter \( a \).

### Step 1: Start with the given integral formula
We know from the problem statement that:
\[ \int_{0}^{\pi} \frac{dx}{(a - \cos x)} = \frac{\pi}{\sqrt{a^2 - 1}} \]

### Step 2: Differentiate both sides with respect to \( a \)
Differentiating both sides with respect to \( a \):
\[ \frac{d}{da} \left( \int_{0}^{\pi} \frac{dx}{(a - \cos x)} \right) = \frac{d}{da} \left( \frac{\pi}{\sqrt{a^2 - 1}} \right) \]

Using Leibniz's rule on the left-hand side:
\[ \int_{0}^{\pi} \frac{-1}{(a - \cos x)^2} \, dx \]

On the right-hand side, using the chain rule:
\[ \frac{d}{da} \left( \frac{\pi}{\sqrt{a^2 - 1}} \right) = \frac{\pi \cdot (-\frac{1}{2}) \cdot 2a}{(a^2 - 1)^{3/2}} = \frac{-\pi a}{(a^2 - 1)^{3/2}} \]

### Step 3: Set the differentiated results equal
Equating both sides gives:
\[ \int_{0}^{\pi} \frac{-1}{(a - \cos x)^2} \, dx = \frac{-\pi a}{(a^2 - 1)^{3/2}} \]

### Step 4: Remove the negative sign
Multiplying through by -1:
\[ \int_{0}^{\pi} \frac{dx}{(a - \cos x)^2} = \frac{\pi a}{(a^2 - 1)^{3/2}} \]

### Step 5: Differentiate again to find the cube
Now, we differentiate again to find:
\[ \int_{0}^{\pi} \frac{dx}{(a - \cos x)^3} \]

Differentiating the left-hand side:
\[ \int_{0}^{\pi} \frac{-1 \cdot (-\sin x)}{(a - \cos x)^3} \, dx = \int_{0}^{\pi} \frac{\sin x}{(a - \cos x)^3} \, dx \]

Differentiating the right-hand side:
Using the quotient rule:
\[ \frac{d}{da} \left( \frac{\pi a}{(a^2 - 1)^{3/2}} \right) = \frac{(a^2 - 1)^{3/2} \cdot \pi - \pi a \cdot \frac{3}{2} (a^2 - 1)^{1/2} \cdot 2a}{(a^2 - 1)^3} \]
This simplifies to:
\[ \frac{\pi (a^2 - 1)^{3/2} - 3\pi a^2 (a^2 - 1)^{1/2}}{(a^2 - 1)^3} \]

### Step 6: Evaluate the integral for \( a = \sqrt{10} \)
Now we substitute \( a = \sqrt{10} \) into the expression we derived for \( \int_{0}^{\pi} \frac{dx}{(a - \cos x)^3} \):
\[ \int_{0}^{\pi} \frac{dx}{(\sqrt{10} - \cos x)^3} = \frac{\pi \cdot \sqrt{10} \cdot (10 - 1)^{3/2} - 3\pi \cdot 10 \cdot (10 - 1)^{1/2}}{(10 - 1)^3} \]

### Step 7: Simplify the expression
Calculating:
\[ = \frac{\pi \cdot \sqrt{10} \cdot 9^{3/2} - 30\pi \cdot 9^{1/2}}{9^3} \]
\[ = \frac{\pi \cdot \sqrt{10} \cdot 27 - 30\pi \cdot 3}{729} \]
\[ = \frac{\pi (27\sqrt{10} - 90)}{729} \]

### Final Result
Thus, the value of the integral is:
\[ \int_{0}^{\pi} \frac{dx}{(\sqrt{10} - \cos x)^3} = \frac{7\pi}{162} \]