Antiderevative

[ferhaouim3]

hi there,

can you find the antiderevative of: square root of (1+cosx), thank you :lol:

 

[opticaltempest]

We want to find the derivative of \(\displaystyle \sqrt {1 + \cos \left( x \right)} \\)

It helps to rewrite the our original expression as, \(\displaystyle \left( {1 + \cos (x)} \right)^{\frac{1}{2}}
\\)


Do you see how you can apply the general power rule?

EDIT: It was late... I read it as derivative... Sorry [/b]

 

[soroban]

Hello, ferhaouim3!

Pay no attention to the man behind the curtain . . .


We have: . \(\displaystyle \L\int \sqrt{1\,+\,\cos(x)}\,dx\)


Recall the identity: .$\displaystyle cos^2\left(\frac{x}{2}\right) \:= \:\frac{1\,+\,\cos(x)}{2}\;\;\Rightarrow\;\;1\,+\,\cos(x)\:=\:2\cdot\cos^2\left(\frac{x}{2}\right)$

. . Hence: .$\displaystyle \sqrt{1\,+\,\cos(x)} \:=\:\sqrt{2}\cdot\cos\left(\frac{x}{2}\right)$


The integral becomes: .\(\displaystyle \L\sqrt{2}\int\cos\left(\frac{x}{2}\right)\,dx\) . . . got it?