Evaluate the integral .
E) integral diverges

Question

Evaluate the integral  .
E) integral diverges

Quizplus · Accepted Answer

The integral is of the form $\int \frac{1}{x^2+a^2}\ dx$, which we can evaluate using the substitution $u = \frac{x}{a}$, $du = \frac{1}{a}\ dx$. Applying this substitution, we get $\int \frac{1}{x^2+a^2}\ dx = \frac{1}{a}\arctan{\frac{x}{a}}+C$. Thus, the integral evaluates to $\frac{1}{2}\left[\frac{1}{\sqrt{5}}\arctan{\frac{x}{\sqrt{5}}}+\frac{1}{\sqrt{13}}\arctan\frac{x}{\sqrt{13}}\right]_{0}^{1} = \frac{1}{2}\left(\frac{1}{\sqrt{5}}\arctan\frac{1}{\sqrt{5}}+\frac{1}{\sqrt{13}}\arctan\frac{1}{\sqrt{13}}\right)$, which matches choice C.

Evaluate the Integral