Evaluate : `intsin4x.e^(tan^(2)x)dx`

Explanation

To evaluate the integral \(\int \sin(4x) e^{\tan^2 x} \, dx\), we can follow these steps:

### Step 1: Rewrite \(\sin(4x)\)
Using the double angle formula for sine, we know that:
\[ \sin(4x) = 2 \sin(2x) \cos(2x) \]
Thus, we can rewrite the integral as:
\[ \int \sin(4x) e^{\tan^2 x} \, dx = \int 2 \sin(2x) \cos(2x) e^{\tan^2 x} \, dx \]

### Step 2: Rewrite \(\sin(2x)\)
Using the double angle formula again:
\[ \sin(2x) = 2 \sin(x) \cos(x) \]
Substituting this into the integral gives:
\[ \int 2 \sin(2x) \cos(2x) e^{\tan^2 x} \, dx = \int 4 \sin(x) \cos(x) \cos(2x) e^{\tan^2 x} \, dx \]

### Step 3: Substitute for \(\tan^2 x\)
Let \(t = \tan^2 x\). Then, differentiating gives:
\[ dt = 2 \tan x \sec^2 x \, dx \quad \Rightarrow \quad dx = \frac{dt}{2 \tan x \sec^2 x} \]
We also know that \(\tan x = \frac{\sin x}{\cos x}\) and \(\sec^2 x = \frac{1}{\cos^2 x}\).

### Step 4: Rewrite the integral in terms of \(t\)
Now we rewrite the integral using the substitution:
\[ \int 4 \sin(x) \cos(x) \cos(2x) e^{t} \frac{dt}{2 \tan x \sec^2 x} \]
This simplifies to:
\[ 2 \int \frac{\sin(x) \cos(x) \cos(2x) e^{t}}{\tan x \sec^2 x} \, dt \]

### Step 5: Simplify the expression
Using the identities:
\[ \sin(x) = \tan x \cos(x) \]
we can simplify:
\[ \int 2 \cos(2x) e^{t} \, dt \]

### Step 6: Integrate
Now we can integrate:
\[ \int e^{t} \, dt = e^{t} + C \]
So we have:
\[ 2 e^{t} + C \]

### Step 7: Substitute back for \(t\)
Substituting back \(t = \tan^2 x\):
\[ 2 e^{\tan^2 x} + C \]

### Step 8: Final expression
Thus, the final result of the integral is:
\[ \int \sin(4x) e^{\tan^2 x} \, dx = 2 e^{\tan^2 x} + C \]