The Notes: `int_0^(pi/2) cos^2(theta) d theta` Evaluate the integral

$\displaystyle{\int_{{0}}^{{\frac{\pi}{{2}}}}}{{\cos}}^{{2}}{\left(\theta\right)}{d}{\left(\theta\right)}$

Use the identity: $\displaystyle{{\cos}}^{{2}}{x}=\frac{{{1}+{\cos{{\left({2}{x}\right)}}}}}{{2}}$

$\displaystyle\int{{\cos}}^{{2}}{\left(\theta\right)}{d}{\left(\theta\right)}=\int\frac{{{1}+{\cos{{\left({2}\theta\right)}}}}}{{2}}{d}\theta$

$\displaystyle=\frac{{1}}{{2}}\int{\left({1}+{\cos{{\left({2}\theta\right)}}}\right)}{d}\theta$

$\displaystyle=\frac{{1}}{{2}}{\left(\int{1}{d}\theta+\int{\cos{{\left({2}\theta\right)}}}{d}\theta\right)}$

$\displaystyle=\frac{{1}}{{2}}{\left(\theta+\frac{{\sin{{\left({2}\theta\right)}}}}{{2}}\right)}$

add a constant C to the solution,

$\displaystyle=\frac{{1}}{{2}}{\left(\theta+\frac{{\sin{{\left({2}\theta\right)}}}}{{2}}\right)}+{C}$

$\displaystyle{\int_{{0}}^{{\frac{\pi}{{2}}}}}{{\cos}}^{{2}}{\left(\theta\right)}{d}\theta={{\left[\frac{{1}}{{2}}{\left(\theta+\frac{{\sin{{\left({2}\theta\right)}}}}{{2}}\right)}\right]}_{{0}}^{{\frac{\pi}{{2}}}}}$

$\displaystyle={\left[\frac{{1}}{{2}}{\left(\frac{\pi}{{2}}+\frac{{\sin{{\left({2}\cdot\frac{\pi}{{2}}\right)}}}}{{2}}\right)}\right]}-{\left[\frac{{1}}{{2}}{\left({0}+\frac{{\sin{{\left({0}\right)}}}}{{2}}\right)}\right]}$