# Thread: Calculus: finding the intergal of sqrt(cos (6x)+1)

1. ## Calculus: finding the intergal of sqrt(cos (6x)+1)

Hello, I would like to know if you can help me with this problem the intergal of sqrt(cos (6x)+1), I’ve understood it as far as cos (1/2 (ax)) but where is the sqrt of 2 coming from?! Thank you.

Find the indefinite integral:

. . . . .$\displaystyle \int\, \bigg(\sqrt{\strut \cos(6x)\, +\, 1\,}\bigg)\, dx$

Step (1)

Apply rule:

. . . . .$\displaystyle \int\, \bigg(\sqrt{\strut \cos(ax)\, +\, 1\,}\bigg)\, dx\, \longrightarrow\, \int\, \bigg(\sqrt{\strut 2\,}\, \cos\left(\dfrac{1}{2}ax\right)\bigg)\, dx$

. . . . .$\displaystyle \int\, \bigg(\sqrt{\strut \cos(6x)\, +\, 1\,}\bigg)\, dx\, =\, \dfrac{1}{3}\, \sqrt{\strut 2\,}\, \sin(3x)$

2. ## Calculus

Hello, I would like to know if you can help me with this problem the intergal of sqrt(cos (6x)+1), I’ve understood it as far as cos (1/2 (ax)) but where is the sqrt of 2 coming from?! Thank you.

3. Originally Posted by kgilyot
Hello, I would like to know if you can help me with this problem the intergal of sqrt(cos (6x)+1), I’ve understood it as far as cos (1/2 (ax)) but where is the sqrt of 2 coming from?! Thank you.

Find the indefinite integral:

. . . . .$\displaystyle \int\, \bigg(\sqrt{\strut \cos(6x)\, +\, 1\,}\bigg)\, dx$

Step (1)

Apply rule:

. . . . .$\displaystyle \int\, \bigg(\sqrt{\strut \cos(ax)\, +\, 1\,}\bigg)\, dx\, \longrightarrow\, \int\, \bigg(\sqrt{\strut 2\,}\, \cos\left(\dfrac{1}{2}ax\right)\bigg)\, dx$

. . . . .$\displaystyle \int\, \bigg(\sqrt{\strut \cos(6x)\, +\, 1\,}\bigg)\, dx\, =\, \dfrac{1}{3}\, \sqrt{\strut 2\,}\, \sin(3x)$
Hint:

cos(2Θ) + 1 = 2*cos2(Θ)