`int tan 2 x tan 3x tan 5x dx = log | sec^(a) 2x. Sec^(b) 3x. Sec^(c) 5x| + k`. Then:\

Options

A. a, b, c are in H.P.
B. a = b = c
C. ab + bc + ca = 0
D. `a^(-3) + b^(-3) + c^(-3) = 3a^(-1) b^(-1) c^(-1)`

Answer

To solve the integral \( \int \tan 2x \tan 3x \tan 5x \, dx \) and express it in the form \( \log | \sec^{a} 2x \cdot \sec^{b} 3x \cdot \sec^{c} 5x | + k \), we will follow these steps:

### Step 1: Rewrite the Integral
We start with the integral:
\[ I = \int \tan 2x \tan 3x \tan 5x \, dx \]

### Step 2: Use the Tangent Addition Formula
We can express \( \tan 5x \) using the tangent addition formula:
\[ \tan 5x = \tan(2x + 3x) = \frac{\tan 2x + \tan 3x}{1 - \tan 2x \tan 3x} \]
This allows us to rewrite the integral in terms of \( \tan 2x \) and \( \tan 3x \).

### Step 3: Substitute into the Integral
Substituting \( \tan 5x \) into the integral gives:
\[ I = \int \tan 2x \tan 3x \left( \frac{\tan 2x + \tan 3x}{1 - \tan 2x \tan 3x} \right) dx \]

### Step 4: Simplify the Integral
This can be simplified to:
\[ I = \int \frac{\tan^2 2x \tan 3x + \tan 2x \tan^2 3x}{1 - \tan 2x \tan 3x} \, dx \]

### Step 5: Integrate Each Term
We can integrate each term separately:
1. The integral of \( \tan^2 2x \tan 3x \) can be computed using the identity \( \tan^2 x = \sec^2 x - 1 \).
2. The integral of \( \tan 2x \tan^2 3x \) can similarly be computed.

### Step 6: Use the Integral of Tangent
Recall that:
\[ \int \tan x \, dx = -\ln |\cos x| = \ln |\sec x| \]
Thus, we can express the integrals of \( \tan 2x \), \( \tan 3x \), and \( \tan 5x \) in terms of logarithms of secants.

### Step 7: Combine the Results
After integrating, we will have:
\[ I = \frac{1}{5} \ln |\sec 5x| - \frac{1}{2} \ln |\sec 2x| - \frac{1}{3} \ln |\sec 3x| + C \]
Using properties of logarithms, we can combine these into a single logarithm:
\[ I = \ln \left| \sec^{\frac{1}{5}} 5x \cdot \sec^{-\frac{1}{2}} 2x \cdot \sec^{-\frac{1}{3}} 3x \right| + C \]

### Step 8: Identify \( a, b, c \)
From the expression, we identify:
- \( a = -\frac{1}{2} \)
- \( b = -\frac{1}{3} \)
- \( c = \frac{1}{5} \)

### Final Answer
Thus, the values of \( a, b, c \) are:
- \( a = -\frac{1}{2} \)
- \( b = -\frac{1}{3} \)
- \( c = \frac{1}{5} \)