부분적분과 치환적분

$\displaystyle \int _{1} ^{2} {} \left( x - 1 \right) f' \left( \frac{x}{2} \right) dx$ $\displaystyle = \left[ \begin{array}{l} \begin{matrix} \\ \end{matrix} 2 \left( x - 1 \right) f \left( \frac{x}{2} \right) \end{array} \right] _{1} ^{2} - \int _{1} ^{2} {} 2 f \left( \frac{x}{2} \right) dx$ $\displaystyle = 2 f \left( 1 \right) - 2 \int _{1} ^{2} {} f \left( \frac{x}{2} \right) dx = 2$ $f ( 1 ) = 4$이므로 $\displaystyle \int _{1} ^{2} {} f \left( \frac{x}{2} \right) dx = 3$ $\displaystyle \frac{x}{2} = t$라 하면 $\displaystyle \frac{1}{2} = \frac{d t}{d x}$ $x = 1$일 때 $\displaystyle t = \frac{1}{2}$, $x = 2$일 때 $t = 1$ $\displaystyle \int _{1} ^{2} {} f \left( \frac{x}{2} \right) dx = 2 \int _{\frac{1}{2}} ^{1} {} f \left( t \right) dt = 3$ 따라서 $\displaystyle \int _{\frac{1}{2}} ^{1} {} f \left( x \right) dx = \frac{3}{2}$