Question

Evaluate the following integral:

\[\int\frac{2 x^2 + 1}{x^2 \left( x^2 + 4 \right)}dx\]

Answer 1

\[\text{Let }I = \int\frac{2 x^2 + 1}{x^2 \left( x^2 + 4 \right)}dx\]
We express
\[\frac{2 x^2 + 1}{x^2 \left( x^2 + 4 \right)} = \frac{A}{x^2} + \frac{B}{x^2 + 4}\]
\[ \Rightarrow 2 x^2 + 1 = A\left( x^2 + 4 \right) + B\left( x^2 \right)\]
Equating the coefficients of `x^2` and constants, we get
\[2 = A + B\text{ and }1 = 4A\]
\[\text{or }A = \frac{1}{4}\text{ and }B = \frac{7}{4}\]
\[ \therefore I = \int\left( \frac{\frac{1}{4}}{x^2} + \frac{\frac{7}{4}}{x^2 + 4} \right)dx\]
\[ = \frac{1}{4}\int\frac{1}{x^2}dx + \frac{7}{4}\int\frac{1}{x^2 + 4} dx\]
\[ = - \frac{1}{4x} + \frac{7}{8} \tan^{- 1} \frac{x}{2} + c\]
\[\text{Hence, }\int\frac{2 x^2 + 1}{x^2 \left( x^2 + 4 \right)}dx = - \frac{1}{4x} + \frac{7}{8} \tan^{- 1} \frac{x}{2} + c\]