Question
Evaluate the following integrals:
$\int\frac{\text{x}^3-3\text{x}^2+5\text{x}-7+\text{x}^2\text{a}^\text{x}}{2\text{x}^2}\text{dx}$
$\int\frac{\text{x}^3-3\text{x}^2+5\text{x}-7+\text{x}^2\text{a}^\text{x}}{2\text{x}^2}\text{dx}$
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Answer
$\int\frac{\text{x}^3-3\text{x}^2+5\text{x}-7+\text{x}^2\text{a}^\text{x}}{2\text{x}^2}\text{dx}$
$=\frac{1}{2}\int\frac{\text{x}^3}{\text{x}^2}\text{dx}-\frac{3}{2}\int\frac{\text{x}^2}{\text{x}^2}\text{dx}+\frac{5}{2}\int\text{x}\frac{\text{x}}{\text{x}^2}\text{dx}-\frac{7}{2}\int\text{x}^{-2}\text{dx}+\frac{1}{2}\int\frac{\text{x}^2\text{a}^\text{x}}{\text{x}^2}\text{dx}$
$=\frac{1}{2}\times\frac{\text{x}^2}{2}-\frac{3}{2}\text{x}+\frac{5}{2}\log\text{x}-\frac{7}{2}\text{x}^{-1}+\frac{1}{2}\frac{\text{a}^\text{x}}{\log\text{a}}+\text{C}$
$=\frac{1}{2}\Big[\frac{\text{x}^2}{2}-3\text{x}+5\log\text{x}-\frac{7}{\text{x}}+\frac{\text{a}^\text{x}}{\log\text{a}}\Big]+\text{C}$
$\therefore\ \int\frac{\text{x}^3-3\text{x}^2+5\text{x}-7+\text{x}^2\text{a}^\text{x}}{2\text{x}^2}\text{dx}$$=\frac{1}{2}\Big[\frac{\text{x}^2}{2}-3\text{x}+5\log\text{x}-\frac{7}{\text{x}}+\frac{\text{a}^\text{x}}{\log\text{a}}\Big]+\text{C}$
$=\frac{1}{2}\int\frac{\text{x}^3}{\text{x}^2}\text{dx}-\frac{3}{2}\int\frac{\text{x}^2}{\text{x}^2}\text{dx}+\frac{5}{2}\int\text{x}\frac{\text{x}}{\text{x}^2}\text{dx}-\frac{7}{2}\int\text{x}^{-2}\text{dx}+\frac{1}{2}\int\frac{\text{x}^2\text{a}^\text{x}}{\text{x}^2}\text{dx}$
$=\frac{1}{2}\times\frac{\text{x}^2}{2}-\frac{3}{2}\text{x}+\frac{5}{2}\log\text{x}-\frac{7}{2}\text{x}^{-1}+\frac{1}{2}\frac{\text{a}^\text{x}}{\log\text{a}}+\text{C}$
$=\frac{1}{2}\Big[\frac{\text{x}^2}{2}-3\text{x}+5\log\text{x}-\frac{7}{\text{x}}+\frac{\text{a}^\text{x}}{\log\text{a}}\Big]+\text{C}$
$\therefore\ \int\frac{\text{x}^3-3\text{x}^2+5\text{x}-7+\text{x}^2\text{a}^\text{x}}{2\text{x}^2}\text{dx}$$=\frac{1}{2}\Big[\frac{\text{x}^2}{2}-3\text{x}+5\log\text{x}-\frac{7}{\text{x}}+\frac{\text{a}^\text{x}}{\log\text{a}}\Big]+\text{C}$
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