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Math Help - Trignometric Integrals

  1. #1
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    Trignometric Integrals

    <br /> <br />
\int \sqrt {1+cos(\frac{t}{2})} dt<br /> <br />

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

    Thank You guys!
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  2. #2
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    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 <br />

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  3. #3
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    thank you soroban
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