Question

Evaluate $\displaystyle \int \frac{dx}{\sqrt{x^2 + 2x + 2}}$.

Hint:

Remember: $\int \frac{dx}{\sqrt{x^2 + a^2}} = \ln|x + \sqrt{x^2 + a^2}| + C$.

Answer 1

Concept: Use completing the square to simplify the expression inside the square root and then apply standard integral formulas. 

Step 1: {Complete the square.}
\[ x^2 + 2x + 2 = (x+1)^2 + 1 \] 
Step 2: {Substitute.}
Let $u = x+1 \Rightarrow du = dx$ \[ \int \frac{dx}{\sqrt{x^2 + 2x + 2}} = \int \frac{du}{\sqrt{u^2 + 1}} \] 
Step 3: {Use standard formula.}
\[ \int \frac{du}{\sqrt{u^2 + 1}} = \ln \left| u + \sqrt{u^2 + 1} \right| + C \] 
Step 4: {Substitute back.}
\[ = \ln \left| x+1 + \sqrt{x^2 + 2x + 2} \right| + C \] 
Step 5: {Conclusion.}
\[ \boxed{\int \frac{dx}{\sqrt{x^2 + 2x + 2}} = \ln \left| x+1 + \sqrt{x^2 + 2x + 2} \right| + C} \]