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