1. **State the problem:** We need to evaluate the integral $$\int \frac{1}{\sqrt{2x^{2}+3x+2}}\,dx$$. 2. **Identify the quadratic expression:** The expression inside the square root is $$2x^{2}+3x+2$$. 3. **Complete the square:** To simplify the integral, we complete the square for the quadratic. $$2x^{2}+3x+2 = 2\left(x^{2}+\frac{3}{2}x\right)+2$$ Inside the parentheses, complete the square: $$x^{2}+\frac{3}{2}x = \left(x+\frac{3}{4}\right)^{2} - \left(\frac{3}{4}\right)^{2} = \left(x+\frac{3}{4}\right)^{2} - \frac{9}{16}$$ So, $$2x^{2}+3x+2 = 2\left[\left(x+\frac{3}{4}\right)^{2} - \frac{9}{16}\right] + 2 = 2\left(x+\frac{3}{4}\right)^{2} - \frac{18}{16} + 2 = 2\left(x+\frac{3}{4}\right)^{2} - \frac{9}{8} + 2$$ Simplify constants: $$-\frac{9}{8} + 2 = -\frac{9}{8} + \frac{16}{8} = \frac{7}{8}$$ Therefore, $$2x^{2}+3x+2 = 2\left(x+\frac{3}{4}\right)^{2} + \frac{7}{8}$$ 4. **Rewrite the integral:** $$\int \frac{1}{\sqrt{2\left(x+\frac{3}{4}\right)^{2} + \frac{7}{8}}} \, dx$$ 5. **Substitute:** Let $$u = x + \frac{3}{4} \implies du = dx$$ The integral becomes: $$\int \frac{1}{\sqrt{2u^{2} + \frac{7}{8}}} \, du$$ 6. **Factor constants inside the square root:** $$\sqrt{2u^{2} + \frac{7}{8}} = \sqrt{2\left(u^{2} + \frac{7}{16}\right)} = \sqrt{2} \sqrt{u^{2} + \frac{7}{16}}$$ So the integral is: $$\int \frac{1}{\sqrt{2} \sqrt{u^{2} + \frac{7}{16}}} \, du = \frac{1}{\sqrt{2}} \int \frac{1}{\sqrt{u^{2} + \left(\frac{\sqrt{7}}{4}\right)^{2}}} \, du$$ 7. **Use the standard integral formula:** $$\int \frac{1}{\sqrt{u^{2} + a^{2}}} \, du = \ln\left| u + \sqrt{u^{2} + a^{2}} \right| + C$$ Here, $$a = \frac{\sqrt{7}}{4}$$. 8. **Apply the formula:** $$\frac{1}{\sqrt{2}} \ln\left| u + \sqrt{u^{2} + \left(\frac{\sqrt{7}}{4}\right)^{2}} \right| + C$$ 9. **Substitute back for $$u$$:** $$\frac{1}{\sqrt{2}} \ln\left| x + \frac{3}{4} + \sqrt{\left(x + \frac{3}{4}\right)^{2} + \frac{7}{16}} \right| + C$$ **Final answer:** $$\boxed{\int \frac{1}{\sqrt{2x^{2}+3x+2}}\,dx = \frac{1}{\sqrt{2}} \ln\left| x + \frac{3}{4} + \sqrt{\left(x + \frac{3}{4}\right)^{2} + \frac{7}{16}} \right| + C}$$