Notes: `int_0^1 cos(pit/2) dt` Evaluate the definite integral.

Sunday, 1 June 2014

$\displaystyle{\int_{{0}}^{{1}}}{\cos{{\left(\pi\frac{{t}}{{2}}\right)}}}{\left.{d}{t}\right.}$ Evaluate the definite integral.

Given: $\displaystyle{\int_{{0}}^{{1}}}{\cos{{\left(\pi\frac{{t}}{{2}}\right)}}}{\left.{d}{t}\right.}$

Integrate using the u-substitution method.

Let $\displaystyle{u}=\frac{\pi}{{2}}{t}$

$\displaystyle\frac{{{d}{u}}}{{\left.{d}{t}}}=\frac{\pi}{{2}}$

$\displaystyle{\left.{d}{t}\right.}=\frac{{2}}{\pi}\cdot{d}{u}$

$\displaystyle={\int_{{0}}^{{1}}}{\cos{{\left({u}\right)}}}\cdot\frac{{2}}{\pi}{d}{u}$

$\displaystyle=\frac{{2}}{\pi}{\int_{{0}}^{{1}}}{\cos{{\left({u}\right)}}}{d}{u}$

$\displaystyle=\frac{{2}}{\pi}{\sin{{\left({u}\right)}}}$ Evaluated from t=0 to t=1.

$\displaystyle=\frac{{2}}{\pi}{\sin{{\left(\frac{\pi}{{2}}{t}\right)}}}$ Evaluated from t=0 to t=1.

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

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

$\displaystyle=\frac{{2}}{\pi}{\left[{1}-{0}\right]}$

$\displaystyle=\frac{{2}}{\pi}$

$\displaystyle={.637}$

Given: $\displaystyle{\int_{{0}}^{{1}}}{\cos{{\left(\pi\frac{{t}}{{2}}\right)}}}{\left.{d}{t}\right.}$

Integrate using the u-substitution method.

Let $\displaystyle{u}=\frac{\pi}{{2}}{t}$

$\displaystyle\frac{{{d}{u}}}{{\left.{d}{t}}}=\frac{\pi}{{2}}$

$\displaystyle{\left.{d}{t}\right.}=\frac{{2}}{\pi}\cdot{d}{u}$

$\displaystyle={\int_{{0}}^{{1}}}{\cos{{\left({u}\right)}}}\cdot\frac{{2}}{\pi}{d}{u}$

$\displaystyle=\frac{{2}}{\pi}{\int_{{0}}^{{1}}}{\cos{{\left({u}\right)}}}{d}{u}$

$\displaystyle=\frac{{2}}{\pi}{\sin{{\left({u}\right)}}}$ Evaluated from t=0 to t=1.

$\displaystyle=\frac{{2}}{\pi}{\sin{{\left(\frac{\pi}{{2}}{t}\right)}}}$ Evaluated from t=0 to t=1.

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

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

$\displaystyle=\frac{{2}}{\pi}{\left[{1}-{0}\right]}$

$\displaystyle=\frac{{2}}{\pi}$

$\displaystyle={.637}$

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