Questions: ∫ dx / (x^2 + 2x + 26)

Solution Steps

To evaluate the integral \(\int \frac{dx}{x^2 + 2x + 26}\), we can complete the square in the denominator and then use a trigonometric substitution to simplify the integral.

To evaluate the integral \(\int \frac{dx}{x^2 + 2x + 26}\), we first complete the square in the denominator: \[ x^2 + 2x + 26 = (x + 1)^2 + 25 \]

Next, we use the trigonometric substitution \(u = x + 1\), which transforms the integral into: \[ \int \frac{du}{u^2 + 25} \]

We recognize that the integral \(\int \frac{du}{u^2 + a^2}\) has the standard result \(\frac{1}{a} \arctan\left(\frac{u}{a}\right)\). Here, \(a = 5\), so we have: \[ \int \frac{du}{u^2 + 25} = \frac{1}{5} \arctan\left(\frac{u}{5}\right) \]

Substituting back \(u = x + 1\), we get: \[ \frac{1}{5} \arctan\left(\frac{x + 1}{5}\right) \]

Final Answer