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Introduction to integration

Examples using integration rules

1. \begin{eqnarray*} 
\int\frac{2x^3 + 3x + 1}{x} \; \mathrm{d} x &=& \int\left(\frac{2x^3}{x} + \frac{3x}{x} + \frac{1}{x}\right) \; \mathrm{d} x\\  
&=& \int \left(2x^2 + 3 + \frac{1}{x}\right) \; \mathrm{d} x\\  
&=& \frac{2x^3}{3} + 3x + \ln x + C 
\end{eqnarray*}
2. \begin{eqnarray*}  
&& \int (\cos x - e^x) \; \mathrm{d} x \\  
&=& \int \cos x \; \mathrm{d} x - \int e^x \; \mathrm{d} x \\  
&=& \sin x - e^x + {C}
\end{eqnarray*}
3. \begin{eqnarray*} 
\int\sqrt{x^3} + x^{\frac{2}{3}} \; \mathrm{d} x&=& \int \left(x^\frac{3}{2} + x^\frac{2}{3}\right)\; \mathrm{d} x\\  
&=& \frac{x^\frac{5}{2}}{\frac{5}{2}} +  
\frac{x^\frac{5}{3}}{\frac{5}{3}} + {C} \\  
&=& \frac{2x^\frac{5}{2}}{5} + \frac{3x^\frac{5}{3}}{5} + {C} 
\end{eqnarray*}


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