Question

\[\int\frac{x^3 - 3 x^2 + 5x - 7 + x^2 a^x}{2 x^2} dx\]

Answer 1

\[\int\left( \frac{x^3 - 3 x^2 + 5x - 7 + x^2 a^x}{2 x^2} \right)dx\]
\[ = \int\left( \frac{x^3}{2 x^2} - \frac{3 x^2}{2 x^2} + \frac{5x}{2 x^2} - \frac{7}{2 x^2} + \frac{x^2 a^x}{2 x^2} \right)dx\]
\[ = \int\left( \frac{x}{2} - \frac{3}{2} + \frac{5}{2x} - \frac{7}{2} x^{- 2} + \frac{a^x}{2} \right)dx\]
\[ = \frac{1}{2}\ \text{∫  x dx} - \frac{3}{2}\  ∫ dx + \frac{5}{2} ∫ \frac{dx}{x} - \frac{7}{2}\int x^{- 2} dx + \frac{1}{2}\int a^x dx\]
\[ = \frac{1}{2}\left[ \frac{x^2}{2} \right] - \frac{3}{2}x + \frac{5}{2}\ln \left| x \right| - \frac{7}{2} \left[ \frac{x^{- 2 + 1}}{- 2 + 1} \right] + \frac{1}{2}\left[ \frac{a^x}{\ln a} \right] + C\]
\[ = \frac{x^2}{4} - \frac{3}{2}x + \frac{5}{2}\ln \left| x \right| + \frac{7}{2x} + \frac{a^x}{2 \ln a} + C\]
\[ = \frac{1}{2}\left[ \frac{x^2}{2} - 3x + 5 \ln \left| x \right| + \frac{7}{x} + \frac{a^x}{\ln a} \right] + C\]