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I'm stuck on integration of the following function:

$$ \int_0^{2\pi} \frac{5}{2}|\sin(2t)| dt$$ I understand the few first steps with substitution, etc., but I can't get the end result which is "10".

Could someone give me a step by step solution for it?

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Draw the picture. We want $4$ times area from $t=0$ to $t=\pi/2$, or alternately $8$ times area from $0$ to $\pi/4$. – André Nicolas May 20 '12 at 20:51
up vote 5 down vote accepted

Note that $$\lvert \sin(2t) \rvert = \begin{cases} \sin(2t) & t\in[0,\pi/2]\\ - \sin(2t) & t\in[\pi/2,\pi]\\ \sin(2t) & t\in[\pi,3\pi/2]\\ -\sin(2t) & t\in[3\pi/2, 2\pi] \end{cases}$$ Also, $$\int_0^{\pi/2} \sin(2t) dt = -\int_{\pi/2}^{\pi} \sin(2t) dt = \int_{\pi}^{3\pi/2} \sin(2t) dt = -\int_{3\pi/2}^{2\pi} \sin(2t) dt$$ Hence, the integral $$I = \int_0^{2 \pi} \frac52 \lvert \sin(2t) \rvert dt = 4 \times \int_0^{\pi/2} \frac52 \sin(2t) dt = 10 \times \int_0^{\pi/2} \sin(2t) dt = 10$$

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FGITW - beat me to it. +1 – mixedmath May 20 '12 at 18:48

We have $$\begin{equation} \int^{t=2 \pi}_{t=0} \frac{5}{2} |\text{sin} 2t | \, dt = \frac{5}{2} \int^{t=2 \pi}_{t=0} |\text{sin} 2t | \, dt \end{equation}$$ Now use the substitution $y = 2t$, $dy = 2dt$. The boundaries shift as follows: if $t = 2\pi$ then $y = 4\pi$ and if $t = 0$ then $y = 0$

We plug this into the integral to get $$\begin{equation} \frac{5}{2} \int^{t=2 \pi}_{t=0} |\text{sin} 2t | \, dt = \frac{5}{2} \int^{y=4 \pi}_{y=0} |\text{sin } y | \, \frac{dy}{2} = \frac{5}{4} \int^{y=4 \pi}_{y=0} |\text{sin } y | \, dy \end{equation}$$

It remains to remember that sin$x \geq 0$ for $x \in [0,\pi]$, and sin$x \leq 0$ when $ x \in [\pi, 2\pi]$. Finally, we can also use that sin$x$ is periodic, which means that sin$(x + 2\pi) = $ sin$x$.

So we get $$\begin{equation} \frac{5}{4} \int^{y=4 \pi}_{y=0} |\text{sin } y | \, dy = 2 \left(\frac{5}{4} \int^{y=2 \pi}_{y=0} |\text{sin } y | \, dy \right) = 2 \left(\frac{5}{4} \int^{y= \pi}_{y=0} \text{sin } y \, dy - \frac{5}{4} \int^{y= 2 \pi}_{y=\pi} \text{sin } y \, dy \right) \end{equation}$$

You can now compute the integral to obtain the desired result.

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