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Just a moment while I analyze the image.The uploaded image contains an integral problem that can be expressed as:

\[
\int \frac{1}{2x^2 + 4x + 5} \, dx
\]

To solve this, we'll first complete the square for the quadratic in the denominator.

### Step 1: Completing the Square
We have:
\[
2x^2 + 4x + 5
\]
Factor out the 2 from the first two terms:
\[
2(x^2 + 2x) + 5
\]
Now, complete the square inside the parentheses:
\[
x^2 + 2x = (x+1)^2 - 1
\]
So we get:
\[
2((x+1)^2 - 1) + 5 = 2(x+1)^2 + 3
\]

### Step 2: Substitute and Solve
Now, the integral becomes:
\[
\int \frac{1}{2(x+1)^2 + 3} \, dx
\]
We can factor out the constant 2 from the denominator:
\[
\frac{1}{2} \int \frac{1}{(x+1)^2 + \frac{3}{2}} \, dx
\]
This is now in the form of a standard arctangent integral:
\[
\int \frac{1}{u^2 + a^2} \, du = \frac{1}{a} \arctan\left( \frac{u}{a} \right)
\]
Here, \( u = x+1 \) and \( a = \sqrt{\frac{3}{2}} \). Applying this, we get:
\[
\frac{1}{\sqrt{\frac{3}{2}}} \arctan\left( \frac{x+1}{\sqrt{\frac{3}{2}}} \right)
\]
Simplifying the constant:
\[
\frac{1}{\sqrt{\frac{3}{2}}} = \frac{\sqrt{2}}{\sqrt{3}}
\]
Thus, the final solution is:
\[
\frac{\sqrt{2}}{\sqrt{3}} \arctan\left( \frac{x+1}{\sqrt{\frac{3}{2}}} \right) + C
\]

Would you like further details on this solution or have any questions?

Here are 5 related questions to explore:
1. How is completing the square used to simplify integrals?
2. What is the formula for the integral of \( \frac{1}{x^2 + a^2} \)?
3. How do you handle integrals involving quadratic expressions in the denominator?
4. What role does substitution play in simplifying complex integrals?
5. Can you generalize this method for other quadratic functions?

**Tip:** Completing the square is a powerful technique to simplify integrals involving quadratics!

Learn how to solve the integral \int \frac{1}{2x^2 + 4x + 5} \, dx using the completing the square method and applying the arctangent integral formula. This solution involves algebraic manipulation and trigonometric substitution, suitable for students in advanced calculus.

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