Examples of Integration

Evaluate
(i) \int {\left(  {3{{\text{x}}^6} - 2{{\text{x}}^2} + 7{\text{x}} + 1} \right)} {\text{ dx}}
(ii) \int  {\frac{{{{\text{t}}^2} - 2{{\text{t}}^4}}}{{{{\text{t}}^4}}}{\text{ dt}}}
Solution:
(i) \int {\left( {3{{\text{x}}^6} - 2{{\text{x}}^2} +  7{\text{x}} + 1} \right)} {\text{ dx}}

\begin{gathered} = 3\int {{{\text{x}}^6}{\text{ dx}} - 2\int {{{\text{x}}^2}{\text{ dx}}  + 7\int {{\text{x dx}} + \int {1{\text{ dx}}} } } } \\ = 3\frac{{{{\text{x}}^{6 + 1}}}}{{6 + 1}} - 2\frac{{{{\text{x}}^{2 +  1}}}}{{2 + 1}} + 7\frac{{{{\text{x}}^{1 + 1}}}}{{1 + 1}} + {\text{x}} + {\text{c}} \\ = \frac{3}{7}{{\text{x}}^7} - \frac{2}{3}{{\text{x}}^3} +  \frac{7}{2}{{\text{x}}^2} + {\text{x}} + {\text{c}} \\ \end{gathered}


(ii) \int {\frac{{{{\text{t}}^2} -  2{{\text{t}}^4}}}{{{{\text{t}}^4}}}{\text{ dt}}}

\begin{gathered} = \int {\left( {\frac{{{{\text{t}}^2}}}{{{{\text{t}}^4}}} -  2\frac{{{{\text{t}}^4}}}{{{{\text{t}}^4}}}} \right)} {\text{ dt}} = \int  {\left( {\frac{{\text{1}}}{{{{\text{t}}^2}}} - 2} \right)} {\text{ dt}} \\ = \int {\left( {{{\text{t}}^{ - 2}} - 2} \right)} {\text{ dt}} = \int  {{{\text{t}}^{ - 2}}{\text{ dt}} - 2\int {1{\text{ dt}}} } \\ = \frac{{{{\text{t}}^{ - 2 + 1}}}}{{ - 2 + 1}} - 2{\text{t}} +  {\text{c}} \\ =  - {{\text{t}}^{ - 1}} -  2{\text{t}} + {\text{c}} \\ =  - \frac{1}{{{{\text{t}}^1}}} -  2{\text{t}} + {\text{c}} \\ \end{gathered}

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