Evaluate: {eq}\displaystyle \int_{\displaystyle 0}^{\frac{\displaystyle \pi}{\displaystyle 2}} \; tan(\frac{\displaystyle x}{\displaystyle 2}) dx {/eq}

Question:

Evaluate:

{eq}\displaystyle \int_{\displaystyle 0}^{\frac{\displaystyle \pi}{\displaystyle 2}} \; tan(\frac{\displaystyle x}{\displaystyle 2}) dx {/eq}

Integration in Calculus:

The definite integral is used to calculate the exact area. We are given a tangent trigonometric function.

To solve this problem, we'll use the trig-ratio {eq}\displaystyle \tan(\frac{\displaystyle x}{\displaystyle 2}) = \dfrac{ \sin(\frac{\displaystyle x}{\displaystyle 2}) }{ \cos(\frac{\displaystyle x}{\displaystyle 2}) } {/eq}

Next, we'll apply u-substitution and the common integral {eq}\displaystyle \int u^n \, \mathrm{d}u=\dfrac{u^{n+1}}{n+1}+C. {/eq}.

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We are given:

{eq}\displaystyle \int_{\displaystyle 0}^{\frac{\displaystyle \pi}{\displaystyle 2}} \; tan(\frac{\displaystyle x}{\displaystyle 2})...