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

Answer

To solve the integral

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

we will follow these steps:

### Step 1: Simplify the expression under the square root
First, we factor out 2 from the expression inside the square root:

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

### Step 2: Rewrite the integral
Now we can rewrite the integral as:

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

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

\[ x^2 + \frac{3}{2}x - 1 = \left(x + \frac{3}{4}\right)^2 - \left(\frac{3}{4}\right)^2 - 1. \]
Calculating \(\left(\frac{3}{4}\right)^2\):

\[ \left(\frac{3}{4}\right)^2 = \frac{9}{16}. \]

Thus, we have:

\[ x^2 + \frac{3}{2}x - 1 = \left(x + \frac{3}{4}\right)^2 - \left(\frac{9}{16} + 1\right) = \left(x + \frac{3}{4}\right)^2 - \frac{25}{16}. \]

### Step 4: Substitute back into the integral
Now we substitute this back into the integral:

\[ \frac{1}{\sqrt{2}} \int \frac{dx}{\sqrt{\left(x + \frac{3}{4}\right)^2 - \left(\frac{5}{4}\right)^2}}. \]

### Step 5: Use a trigonometric substitution
We can use the substitution \(x + \frac{3}{4} = \frac{5}{4} \sec(\theta)\). Then, \(dx = \frac{5}{4} \sec(\theta) \tan(\theta) d\theta\).

Substituting these into the integral gives:

\[ \frac{1}{\sqrt{2}} \int \frac{\frac{5}{4} \sec(\theta) \tan(\theta) d\theta}{\sqrt{\left(\frac{5}{4}\right)^2 \sec^2(\theta) - \left(\frac{5}{4}\right)^2}} = \frac{1}{\sqrt{2}} \int \frac{\frac{5}{4} \sec(\theta) \tan(\theta) d\theta}{\frac{5}{4} \sqrt{\sec^2(\theta) - 1}}. \]

### Step 6: Simplify the integral
This simplifies to:

\[ \frac{1}{\sqrt{2}} \cdot \frac{5}{4} \int d\theta = \frac{5}{4\sqrt{2}} \theta + C. \]

### Step 7: Back substitute
Now we need to back substitute for \(\theta\):

\[ \theta = \sec^{-1}\left(\frac{4}{5}(x + \frac{3}{4})\right). \]

Thus, the final answer is:

\[ \frac{5}{4\sqrt{2}} \sec^{-1}\left(\frac{4}{5}(x + \frac{3}{4})\right) + C. \]