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Introduction to integration

Solutions to the examples using definite integrals

  1.  \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*}
  2.  \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*}

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