Question

Evaluate : \( \int_0^2 \frac{1}{\sqrt{x^2 + 2x + 3}} \, dx \).

Hint:

Always simplify the quadratic denominator by completing the square first. This transforms the integral into one of the nine standard forms.
Check that the coefficient of \( x^2 \) is \( +1 \) before completing the square to avoid sign errors inside the square root.

Answer 1

Concept:
• Integration of irrational algebraic functions involving a quadratic denominator \( \sqrt{ax^2 + bx + c} \) often requires completing the square.
• Standard Integral Formula: \( \int \frac{dx}{\sqrt{x^2 + a^2}} = \log |x + \sqrt{x^2 + a^2}| + C \).
• Fundamental Theorem of Calculus: \( \int_a^b f(x) \, dx = F(b) - F(a) \).

Step 1: Complete the square for the quadratic expression in the denominator
The quadratic expression is \( x^2 + 2x + 3 \).
To complete the square, we take half of the coefficient of \( x \), which is \( 1 \), and square it:
\[ x^2 + 2x + 3 = (x^2 + 2x + 1) + 2 \]
Writing it in terms of squares:
\[ = (x + 1)^2 + (\sqrt{2})^2 \]

Step 2: Substitute back into the integral and apply the formula
Let the integral be \( I \):
\[ I = \int_0^2 \frac{1}{\sqrt{(x + 1)^2 + (\sqrt{2})^2}} \, dx \]
Using the formula \( \int \frac{dx}{\sqrt{X^2 + A^2}} = \log |X + \sqrt{X^2 + A^2}| \), where \( X = x+1 \) and \( A = \sqrt{2} \):
\[ I = \left[ \log \left| (x + 1) + \sqrt{(x + 1)^2 + (\sqrt{2})^2} \right| \right]_0^2 \]
Simplify the expression inside the log back to its original quadratic form:
\[ I = \left[ \log \left| (x + 1) + \sqrt{x^2 + 2x + 3} \right| \right]_0^2 \]

Step 3: Evaluate the definite integral using the limits
Substitute the upper limit \( x = 2 \):
\[ \text{Value at upper limit} = \log |(2 + 1) + \sqrt{2^2 + 2(2) + 3}| \]
\[ = \log |3 + \sqrt{4 + 4 + 3}| = \log(3 + \sqrt{11}) \]
Substitute the lower limit \( x = 0 \):
\[ \text{Value at lower limit} = \log |(0 + 1) + \sqrt{0^2 + 2(0) + 3}| \]
\[ = \log |1 + \sqrt{3}| = \log(1 + \sqrt{3}) \]
Subtract the lower limit value from the upper limit value:
\[ I = \log(3 + \sqrt{11}) - \log(1 + \sqrt{3}) \]
Using the property \( \log A - \log B = \log(A/B) \):
\[ I = \log \left( \frac{3 + \sqrt{11}}{1 + \sqrt{3}} \right) \]