`int(dx)/(sqrt(1+2x-3x^(2)))`

Explanation

To solve the integral

\[ \int \frac{dx}{\sqrt{1 + 2x - 3x^2}}, \]

we will follow these steps:

### Step 1: Factor out the coefficient of \(x^2\)

The expression under the square root is \(1 + 2x - 3x^2\). We can factor out \(-3\) from the quadratic expression.

\[ 1 + 2x - 3x^2 = -3\left(\frac{-1}{3} - \frac{2}{3}x + x^2\right). \]

### Step 2: Rewrite the integral

Now, we can rewrite the integral by factoring out \(-3\):

\[ \int \frac{dx}{\sqrt{1 + 2x - 3x^2}} = \int \frac{dx}{\sqrt{-3\left(x^2 + \frac{2}{3}x - \frac{1}{3}\right)}}. \]

This simplifies to:

\[ \frac{1}{\sqrt{3}} \int \frac{dx}{\sqrt{-\left(x^2 + \frac{2}{3}x - \frac{1}{3}\right)}}. \]

### Step 3: Complete the square

Next, we complete the square for the quadratic expression \(x^2 + \frac{2}{3}x - \frac{1}{3}\):

\[ x^2 + \frac{2}{3}x - \frac{1}{3} = \left(x + \frac{1}{3}\right)^2 - \frac{1}{3} - \frac{1}{3} = \left(x + \frac{1}{3}\right)^2 - \frac{2}{3}. \]

### Step 4: Substitute into the integral

Now we substitute this back into the integral:

\[ \frac{1}{\sqrt{3}} \int \frac{dx}{\sqrt{-\left(\left(x + \frac{1}{3}\right)^2 - \frac{2}{3}\right)}}. \]

### Step 5: Use trigonometric substitution

We can use the substitution \(x + \frac{1}{3} = \sqrt{\frac{2}{3}} \sin \theta\):

\[ dx = \sqrt{\frac{2}{3}} \cos \theta \, d\theta. \]

Now, substitute this into the integral:

\[ \frac{1}{\sqrt{3}} \int \frac{\sqrt{\frac{2}{3}} \cos \theta \, d\theta}{\sqrt{-\left(\frac{2}{3} \sin^2 \theta - \frac{2}{3}\right)}}. \]

This simplifies to:

\[ \frac{1}{\sqrt{3}} \int \frac{\sqrt{\frac{2}{3}} \cos \theta \, d\theta}{\sqrt{\frac{2}{3}(1 - \sin^2 \theta)}} = \frac{1}{\sqrt{3}} \int \frac{\sqrt{\frac{2}{3}} \cos \theta \, d\theta}{\sqrt{\frac{2}{3} \cos^2 \theta}}. \]

### Step 6: Simplify the integral

This simplifies to:

\[ \frac{1}{\sqrt{3}} \int d\theta = \frac{1}{\sqrt{3}} \theta + C. \]

### Step 7: Back substitute

Now we need to back substitute for \(\theta\):

\[ \theta = \sin^{-1}\left(\frac{3x + 1}{\sqrt{2}}\right). \]

Thus, the final result is:

\[ \frac{1}{\sqrt{3}} \sin^{-1}\left(\frac{3x + 1}{\sqrt{2}}\right) + C. \]

### Summary of Steps:
1. Factor out the coefficient of \(x^2\).
2. Rewrite the integral.
3. Complete the square.
4. Substitute into the integral.
5. Use trigonometric substitution.
6. Simplify the integral.
7. Back substitute to find the final result.