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Introduction to integration
Solutions to the examples using definite integrals
-
\begin{eqnarray*}
\int^4_1\left(\frac{1}{x} + \frac{1}{\sqrt{x}}\right)\; \mathrm{d} x &=&
\int^4_1\left(\frac{1}{x} + x^{-\frac{1}{2}}\right)\; \mathrm{d} x \\
&=& \bigg[\ln x + 2\sqrt{x}\bigg]^4_1\\
&=& (\ln 4 + 2\sqrt{4}) - (\ln 1 + 2\sqrt{1})\\
&\approx& (1.386 + 4) - (0 + 2) \\
&=& 3.386
\end{eqnarray*}
- \begin{eqnarray*}
&& \int^{\pi/2}_0(\cos \theta + \sin \theta)\; \mathrm{d}\theta \\ &=&
\bigg[\sin \theta - \cos \theta \bigg]^{\pi/2}_0\\
&=& \left(\sin \frac{\pi}{2} - \cos
\frac{\pi}{2}\right)- (\sin 0 - \cos 0)\\
&=& (1 - 0) - (0 - 1)\\
&=& 2
\end{eqnarray*}