`int_(pi/6)^(pi/3) tan^3xsin^2 3x(2sec^2x sin^2 3x+3tanx.sin6x)dx`

Options

A. `-1/36`
B. `-1/72`
C. `-1/18`
D. `1/36`

Answer

To solve the integral

\[ I = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \tan^3 x \sin^2(3x) \left(2 \sec^2 x \sin^2(3x) + 3 \tan x \sin(6x)\right) dx, \]

we can follow these steps:

### Step 1: Simplifying the Integral
We start by rewriting the integral \(I\):

\[ I = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \tan^3 x \sin^2(3x) \left(2 \sec^2 x \sin^2(3x) + 3 \tan x \sin(6x)\right) dx. \]

### Step 2: Distributing the Terms
Distributing the terms inside the integral gives us:

\[ I = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \left(2 \tan^3 x \sec^2 x \sin^4(3x) + 3 \tan^4 x \sin^2(3x) \sin(6x)\right) dx. \]

### Step 3: Using Trigonometric Identities
Recall that \(\sin(6x) = 2 \sin(3x) \cos(3x)\). We can substitute this into the integral:

\[ I = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \left(2 \tan^3 x \sec^2 x \sin^4(3x) + 6 \tan^4 x \sin^2(3x) \cos(3x)\right) dx. \]

### Step 4: Factor Out Common Terms
We can factor out common terms from the integral:

\[ I = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \sin^2(3x) \left(2 \tan^3 x \sec^2 x \sin^2(3x) + 6 \tan^4 x \cos(3x)\right) dx. \]

### Step 5: Substituting for \(\tan x\) and \(\sec^2 x\)
Using the identities \(\tan x = \frac{\sin x}{\cos x}\) and \(\sec^2 x = 1 + \tan^2 x\), we can rewrite the integral in terms of sine and cosine:

\[ I = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \sin^2(3x) \left(2 \frac{\sin^3 x}{\cos^3 x} (1 + \tan^2 x) \sin^2(3x) + 6 \frac{\sin^4 x}{\cos^4 x} \cos(3x)\right) dx. \]

### Step 6: Evaluating the Integral
Now we can evaluate the integral using numerical methods or further simplifications. However, we can also apply integration techniques such as integration by parts or substitution if necessary.

### Step 7: Applying Limits
After evaluating the integral, we will apply the limits from \(\frac{\pi}{6}\) to \(\frac{\pi}{3}\).

### Step 8: Final Calculation
Finally, we will compute the values at the limits and simplify to find the answer.