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