Question

\[\int\frac{a x^3 + bx}{x^4 + c^2} dx\]

Answer 1

\[\int\frac{\left( a x^3 + bx \right)}{x^4 + c^2}dx\]
\[ = \int\frac{a x^3}{x^4 + c^2}dx + \int\frac{bx}{\left( x^2 \right)^2 + c^2}dx\]
\[ = I_1 + I_2 \left(\text{ say } \right)\]
\[Where\]
\[ I_1 = \int \frac{a x^3}{x^4 + c^2}dx\ \text{and}\ I_2 = \int\frac{bx}{\left( x^2 \right)^2 + c^2}dx\]
\[Now, I_1 = \int\frac{a x^3}{x^4 + c^2}dx\]
\[\text{ let x }^4 + c^2 = t\]
\[ \Rightarrow 4 x^3 dx = dt\]
\[ \Rightarrow x^3 dx = \frac{dt}{4}\]
\[ I_1 = \frac{a}{4}\int\frac{dt}{t}\]
\[ = \frac{a}{4} \text{ log }\left| t \right| + C_1 \]
\[ = \frac{a}{4} \text{ log }\left| x^4 + c^2 \right| + C_1 \]
\[Now, I_2 = \int\frac{bx}{\left( x^2 \right)^2 + c^2}dx\]
\[\text{ let x} ^2 = p\]
\[ \Rightarrow \text{ 2x dx } = dp\]
\[ \Rightarrow \text { x dx }= \frac{dp}{2}\]