Question

\[\int\frac{( x^2 + 1) ( x^2 + 4)}{( x^2 + 3) ( x^2 - 5)} dx\]

Answer 1

\[I=\int\frac{( x^2 + 1) ( x^2 + 4)}{( x^2 + 3) ( x^2 - 5)} dx\]

Since,

\[\frac{\left( x^2 + 1 \right)\left( x^2 + 4 \right)}{\left( x^2 + 3 \right)\left( x^2 - 5 \right)} = \frac{\left[ \left( x^2 + 3 \right) - 2 \right]\left[ \left( x^2 - 5 \right) + 9 \right]}{\left( x^2 + 3 \right)\left( x^2 - 5 \right)}\]
\[ \Rightarrow \frac{\left( x^2 + 1 \right)\left( x^2 + 4 \right)}{\left( x^2 + 3 \right)\left( x^2 - 5 \right)} = \frac{\left( x^2 + 3 \right)\left( x^2 - 5 \right) + 9\left( x^2 + 3 \right) - 2\left( x^2 - 5 \right) - 18}{\left( x^2 + 3 \right)\left( x^2 - 5 \right)}\]

\[\Rightarrow \frac{\left( x^2 + 1 \right)\left( x^2 + 4 \right)}{\left( x^2 + 3 \right)\left( x^2 - 5 \right)} = 1 + \frac{9}{\left( x^2 - 5 \right)} - \frac{2}{\left( x^2 + 3 \right)} - \frac{18}{\left( x^2 + 3 \right)\left( x^2 - 5 \right)}.............(1)\]

Let 

\[I_1 = \int\frac{1}{( x^2 + 3)( x^2 - 5)}\text{ and }x^2 = y\]

\[\Rightarrow \frac{1}{\left( y + 3 \right)\left( y - 5 \right)} = \frac{A}{\left( y + 3 \right)} + \frac{B}{\left( y - 5 \right)}\]
\[ = \frac{A\left( y - 5 \right) + B\left( y + 3 \right)}{\left( y + 3 \right)\left( y - 5 \right)}\]
\[ \Rightarrow \frac{1}{\left( y + 3 \right)\left( y - 5 \right)} = \frac{\left( A + B \right)y - \left( 5A + 3B \right)}{\left( y + 3 \right)\left( y - 5 \right)}\]

Comparing coefficients, we get

\[A + B = 0\text{ and }5A + 3B = - 1\]
\[\text{By solving the equations, we get}\]
\[A = - \frac{1}{8}\text{ and }B = \frac{1}{8}\]

From (1), we get

\[I = \int\left[ 1 + \frac{9}{\left( x^2 - 5 \right)} - \frac{2}{\left( x^2 + 3 \right)} - 18\left( \frac{- 1}{8\left( x^2 + 3 \right)} + \frac{1}{8\left( x^2 - 5 \right)} \right) \right]dx\]

\[\Rightarrow I = \int\left[ 1 + \frac{27}{4\left( x^2 - 5 \right)} + \frac{1}{\left( x^2 + 3 \right)} \right]dx\]
\[ \Rightarrow I = \int1dx + \int\frac{27}{4\left( x^2 - 5 \right)}dx + \int\frac{1}{\left( x^2 + 3 \right)}dx\]
\[ \therefore I = x + \frac{27}{8\sqrt{5}}\ln\left( \left| \frac{x - \sqrt{5}}{x + \sqrt{5}} \right| \right) + \frac{1}{4\sqrt{3}} \tan^{- 1} \left( \frac{x}{\sqrt{3}} \right) + c\]