Trignometric Integrals

**larryboi7** · February 6th 2010, 03:12 PM

$ \int \sqrt {1+cos(\frac{t}{2})} dt $

Can someone show me the steps to integrate this problem thoroughly?

Thank You guys!

**Soroban** · February 6th 2010, 03:45 PM

Hello, larryboi7!

$\int \sqrt {1+\cos\left(\frac{t}{2}\right)}\,dt$

Recall the double-angle identity: . $\cos^2\theta \:=\:\frac{1+\cos2\theta}{2} \quad\Rightarrow\quad 1 + \cos2\theta \:=\:2\cos^2\theta$

So we have: . $1 + \cos\!\left(\frac{t}{2}\right) \;=\;2\cos^2\!\left(\frac{t}{4}\right)$

. . Then: . $\sqrt{1 + \cos\left(\frac{t}{2}\right)} \;=\;\sqrt{2\cos^2\!\left(\frac{t}{4}\right)} \;=\;\sqrt{2}\cos\!\left(\frac{t}{4}\right)$

The integral becomes: . $\sqrt{2}\!\int\cos\!\left(\frac{t}{4}\right) dt \;\;=\;\; 4\sqrt{2}\cdot\sin\left(\frac{t}{4}\right) + C $

**larryboi7** · February 6th 2010, 05:11 PM

thank you soroban

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