# Math Help - Trignometric Integrals

1. ## Trignometric Integrals

$

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

$

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

Thank You guys!

2. 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
$

3. thank you soroban